7: Fourier Transforms: Convolution and Parseval’s Theorem • Multiplication of Signals • Multiplication Example • Convolution Theorem • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum 7: Fourier Transforms: • Summary Convolution and Parseval's Theorem
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 1 / 10 Multiplication of Signals
7: Fourier Transforms: Convolution and Parseval’s Question: What is the Fourier transform of w(t) = u(t)v(t) ? Theorem • Multiplication of Signals • Multiplication Example • Convolution Theorem • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 2 / 10 Multiplication of Signals
7: Fourier Transforms: Convolution and Parseval’s Question: What is the Fourier transform of w(t) = u(t)v(t) ? Theorem +∞ i2πht +∞ i2πgt • Multiplication of Signals Let u(t) = U(h)e dh and v(t) = V (g)e dg • Multiplication Example h=−∞ g=−∞ • Convolution Theorem • Convolution Example R R • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 2 / 10 Multiplication of Signals
7: Fourier Transforms: Convolution and Parseval’s Question: What is the Fourier transform of w(t) = u(t)v(t) ? Theorem +∞ i2πht +∞ i2πgt • Multiplication of Signals Let u(t) = U(h)e dh and v(t) = V (g)e dg • Multiplication Example h=−∞ g=−∞ • Convolution Theorem • Convolution Example R R • Convolution Properties w(t) = u(t)v(t) • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 2 / 10 Multiplication of Signals
7: Fourier Transforms: Convolution and Parseval’s Question: What is the Fourier transform of w(t) = u(t)v(t) ? Theorem +∞ i2πht +∞ i2πgt • Multiplication of Signals Let u(t) = U(h)e dh and v(t) = V (g)e dg • Multiplication Example h=−∞ g=−∞ • Convolution Theorem [Note use of different dummy variables] • Convolution Example R R • Convolution Properties w(t) = u(t)v(t) • Parseval’s Theorem +∞ +∞ • Energy Conservation = U(h)ei2πhtdh V (g)ei2πgtdg • Energy Spectrum h=−∞ g=−∞ • Summary R R
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 2 / 10 Multiplication of Signals
7: Fourier Transforms: Convolution and Parseval’s Question: What is the Fourier transform of w(t) = u(t)v(t) ? Theorem +∞ i2πht +∞ i2πgt • Multiplication of Signals Let u(t) = U(h)e dh and v(t) = V (g)e dg • Multiplication Example h=−∞ g=−∞ • Convolution Theorem [Note use of different dummy variables] • Convolution Example R R • Convolution Properties w(t) = u(t)v(t) • Parseval’s Theorem +∞ +∞ • Energy Conservation = U(h)ei2πhtdh V (g)ei2πgtdg • Energy Spectrum h=−∞ g=−∞ • Summary +∞ +∞ i2π(h+g)t [merge (··· )] = Rh=−∞ U(h) g=−∞ VR(g)e dg dh e R R
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 2 / 10 Multiplication of Signals
7: Fourier Transforms: Convolution and Parseval’s Question: What is the Fourier transform of w(t) = u(t)v(t) ? Theorem +∞ i2πht +∞ i2πgt • Multiplication of Signals Let u(t) = U(h)e dh and v(t) = V (g)e dg • Multiplication Example h=−∞ g=−∞ • Convolution Theorem [Note use of different dummy variables] • Convolution Example R R • Convolution Properties w(t) = u(t)v(t) • Parseval’s Theorem +∞ +∞ • Energy Conservation = U(h)ei2πhtdh V (g)ei2πgtdg • Energy Spectrum h=−∞ g=−∞ • Summary +∞ +∞ i2π(h+g)t [merge (··· )] = Rh=−∞ U(h) g=−∞ VR(g)e dg dh e Now we makeR a change ofR variable in the second integral: g = f − h +∞ +∞ i2πft = h=−∞ U(h) f=−∞ V (f − h)e df dh R R
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 2 / 10 Multiplication of Signals
7: Fourier Transforms: Convolution and Parseval’s Question: What is the Fourier transform of w(t) = u(t)v(t) ? Theorem +∞ i2πht +∞ i2πgt • Multiplication of Signals Let u(t) = U(h)e dh and v(t) = V (g)e dg • Multiplication Example h=−∞ g=−∞ • Convolution Theorem [Note use of different dummy variables] • Convolution Example R R • Convolution Properties w(t) = u(t)v(t) • Parseval’s Theorem +∞ +∞ • Energy Conservation = U(h)ei2πhtdh V (g)ei2πgtdg • Energy Spectrum h=−∞ g=−∞ • Summary +∞ +∞ i2π(h+g)t [merge (··· )] = Rh=−∞ U(h) g=−∞ VR(g)e dg dh e Now we makeR a change ofR variable in the second integral: g = f − h +∞ +∞ i2πft = h=−∞ U(h) f=−∞ V (f − h)e df dh ∞ +∞ i2πft [swap ] = Rf=−∞ h=−∞R U(h)V (f − h)e dh df R R R
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 2 / 10 Multiplication of Signals
7: Fourier Transforms: Convolution and Parseval’s Question: What is the Fourier transform of w(t) = u(t)v(t) ? Theorem +∞ i2πht +∞ i2πgt • Multiplication of Signals Let u(t) = U(h)e dh and v(t) = V (g)e dg • Multiplication Example h=−∞ g=−∞ • Convolution Theorem [Note use of different dummy variables] • Convolution Example R R • Convolution Properties w(t) = u(t)v(t) • Parseval’s Theorem +∞ +∞ • Energy Conservation = U(h)ei2πhtdh V (g)ei2πgtdg • Energy Spectrum h=−∞ g=−∞ • Summary +∞ +∞ i2π(h+g)t [merge (··· )] = Rh=−∞ U(h) g=−∞ VR(g)e dg dh e Now we makeR a change ofR variable in the second integral: g = f − h +∞ +∞ i2πft = h=−∞ U(h) f=−∞ V (f − h)e df dh ∞ +∞ i2πft [swap ] = Rf=−∞ h=−∞R U(h)V (f − h)e dh df +∞ i2πft = Rf=−∞ RW (f)e df R R
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 2 / 10 Multiplication of Signals
7: Fourier Transforms: Convolution and Parseval’s Question: What is the Fourier transform of w(t) = u(t)v(t) ? Theorem +∞ i2πht +∞ i2πgt • Multiplication of Signals Let u(t) = U(h)e dh and v(t) = V (g)e dg • Multiplication Example h=−∞ g=−∞ • Convolution Theorem [Note use of different dummy variables] • Convolution Example R R • Convolution Properties w(t) = u(t)v(t) • Parseval’s Theorem +∞ +∞ • Energy Conservation = U(h)ei2πhtdh V (g)ei2πgtdg • Energy Spectrum h=−∞ g=−∞ • Summary +∞ +∞ i2π(h+g)t [merge (··· )] = Rh=−∞ U(h) g=−∞ VR(g)e dg dh e Now we makeR a change ofR variable in the second integral: g = f − h +∞ +∞ i2πft = h=−∞ U(h) f=−∞ V (f − h)e df dh ∞ +∞ i2πft [swap ] = Rf=−∞ h=−∞R U(h)V (f − h)e dh df +∞ i2πft = Rf=−∞ RW (f)e df R +∞ where W (fR) = h=−∞ U(h)V (f − h)dh R
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 2 / 10 Multiplication of Signals
7: Fourier Transforms: Convolution and Parseval’s Question: What is the Fourier transform of w(t) = u(t)v(t) ? Theorem +∞ i2πht +∞ i2πgt • Multiplication of Signals Let u(t) = U(h)e dh and v(t) = V (g)e dg • Multiplication Example h=−∞ g=−∞ • Convolution Theorem [Note use of different dummy variables] • Convolution Example R R • Convolution Properties w(t) = u(t)v(t) • Parseval’s Theorem +∞ +∞ • Energy Conservation = U(h)ei2πhtdh V (g)ei2πgtdg • Energy Spectrum h=−∞ g=−∞ • Summary +∞ +∞ i2π(h+g)t [merge (··· )] = Rh=−∞ U(h) g=−∞ VR(g)e dg dh e Now we makeR a change ofR variable in the second integral: g = f − h +∞ +∞ i2πft = h=−∞ U(h) f=−∞ V (f − h)e df dh ∞ +∞ i2πft [swap ] = Rf=−∞ h=−∞R U(h)V (f − h)e dh df +∞ i2πft = Rf=−∞ RW (f)e df R +∞ where , W (fR) = h=−∞ U(h)V (f − h)dh U(f) ∗ V (f) R
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 2 / 10 Multiplication of Signals
7: Fourier Transforms: Convolution and Parseval’s Question: What is the Fourier transform of w(t) = u(t)v(t) ? Theorem +∞ i2πht +∞ i2πgt • Multiplication of Signals Let u(t) = U(h)e dh and v(t) = V (g)e dg • Multiplication Example h=−∞ g=−∞ • Convolution Theorem [Note use of different dummy variables] • Convolution Example R R • Convolution Properties w(t) = u(t)v(t) • Parseval’s Theorem +∞ +∞ • Energy Conservation = U(h)ei2πhtdh V (g)ei2πgtdg • Energy Spectrum h=−∞ g=−∞ • Summary +∞ +∞ i2π(h+g)t [merge (··· )] = Rh=−∞ U(h) g=−∞ VR(g)e dg dh e Now we makeR a change ofR variable in the second integral: g = f − h +∞ +∞ i2πft = h=−∞ U(h) f=−∞ V (f − h)e df dh ∞ +∞ i2πft [swap ] = Rf=−∞ h=−∞R U(h)V (f − h)e dh df +∞ i2πft = Rf=−∞ RW (f)e df R +∞ where , W (fR) = h=−∞ U(h)V (f − h)dh U(f) ∗ V (f) This is the convolutionR of the two spectra U(f) and V (f).
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 2 / 10 Multiplication of Signals
7: Fourier Transforms: Convolution and Parseval’s Question: What is the Fourier transform of w(t) = u(t)v(t) ? Theorem +∞ i2πht +∞ i2πgt • Multiplication of Signals Let u(t) = U(h)e dh and v(t) = V (g)e dg • Multiplication Example h=−∞ g=−∞ • Convolution Theorem [Note use of different dummy variables] • Convolution Example R R • Convolution Properties w(t) = u(t)v(t) • Parseval’s Theorem +∞ +∞ • Energy Conservation = U(h)ei2πhtdh V (g)ei2πgtdg • Energy Spectrum h=−∞ g=−∞ • Summary +∞ +∞ i2π(h+g)t [merge (··· )] = Rh=−∞ U(h) g=−∞ VR(g)e dg dh e Now we makeR a change ofR variable in the second integral: g = f − h +∞ +∞ i2πft = h=−∞ U(h) f=−∞ V (f − h)e df dh ∞ +∞ i2πft [swap ] = Rf=−∞ h=−∞R U(h)V (f − h)e dh df +∞ i2πft = Rf=−∞ RW (f)e df R +∞ where , W (fR) = h=−∞ U(h)V (f − h)dh U(f) ∗ V (f) This is the convolutionR of the two spectra U(f) and V (f). w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f)
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 2 / 10 < <
Multiplication Example
7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5
e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( • Convolution Theorem • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 < <
Multiplication Example
7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5
e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( • Convolution Theorem 1 • Convolution Example U(f) = a+i2πf [from before] • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 < <
Multiplication Example
7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5
e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( 0.5 0.4 • Convolution Theorem 0.3 0.2 1 |U(f)| • Convolution Example U(f) = [from before] 0.1 • a+i2πf -5 0 5 Convolution Properties Frequency (Hz) • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 <
Multiplication Example
7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5
e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( 0.5 0.4 • Convolution Theorem 0.3 0.2 1 |U(f)| • Convolution Example U(f) = [from before] 0.1 • a+i2πf -5 0 5 Convolution Properties Frequency (Hz) • Parseval’s Theorem 0.5 • Energy Conservation 0 • Energy Spectrum -0.5
E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 < Multiplication Example 7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5 e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( 0.5 0.4 • Convolution Theorem 0.3 0.2 1 |U(f)| • Convolution Example U(f) = [from before] 0.1 • a+i2πf -5 0 5 Convolution Properties Frequency (Hz) • Parseval’s Theorem 0.5 • Energy Conservation v(t) = cos 2πFt 0 • Energy Spectrum -0.5 E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 < Multiplication Example 7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5 e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( 0.5 0.4 • Convolution Theorem 0.3 0.2 1 |U(f)| • Convolution Example U(f) = [from before] 0.1 • a+i2πf -5 0 5 Convolution Properties Frequency (Hz) • Parseval’s Theorem 0.5 • Energy Conservation v(t) = cos 2πFt 0 • Energy Spectrum -0.5 E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 < Multiplication Example 7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5 e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( 0.5 0.4 • Convolution Theorem 0.3 0.2 1 |U(f)| • Convolution Example U(f) = [from before] 0.1 • a+i2πf -5 0 5 Convolution Properties Frequency (Hz) • Parseval’s Theorem 0.5 • Energy Conservation v(t) = cos 2πFt 0 • Energy Spectrum -0.5 0.4 F=2 0.5 0.5 0.2 |V(f)| 0 -5 0 5 Frequency (Hz) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 < Multiplication Example 7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5 e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( 0.5 0.4 • Convolution Theorem 0.3 0.2 1 |U(f)| • Convolution Example U(f) = [from before] 0.1 • a+i2πf -5 0 5 Convolution Properties Frequency (Hz) • Parseval’s Theorem 0.5 • Energy Conservation v(t) = cos 2πFt 0 • Energy Spectrum -0.5 0.4 F=2 0.5 0.5 0.2 |V(f)| 0 -5 0 5 Frequency (Hz) 1 a=2, F=2 0.5 w(t) 0 -0.5 -5 0 5 Time (s) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 < Multiplication Example 7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5 e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( 0.5 0.4 • Convolution Theorem 0.3 0.2 1 |U(f)| • Convolution Example U(f) = [from before] 0.1 • a+i2πf -5 0 5 Convolution Properties Frequency (Hz) • Parseval’s Theorem 0.5 • Energy Conservation v(t) = cos 2πFt 0 • Energy Spectrum -0.5 0.4 F=2 0.5 0.5 W f U f V f 0.2 ( ) = ( ) ∗ ( ) |V(f)| 0 -5 0 5 Frequency (Hz) 1 a=2, F=2 0.5 w(t) 0 -0.5 -5 0 5 Time (s) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 < Multiplication Example 7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5 e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( 0.5 0.4 • Convolution Theorem 0.3 0.2 1 |U(f)| • Convolution Example U(f) = [from before] 0.1 • a+i2πf -5 0 5 Convolution Properties Frequency (Hz) • Parseval’s Theorem 0.5 • Energy Conservation v(t) = cos 2πFt 0 • Energy Spectrum -0.5 0.4 F=2 0.5 0.5 W f U f V f 0.2 ( ) = ( ) ∗ ( ) |V(f)| 0 0.5 0.5 -5 0 5 = a+i2π(f+F ) + a+i2π(f−F ) Frequency (Hz) 1 a=2, F=2 0.5 w(t) 0 -0.5 -5 0 5 Time (s) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 < Multiplication Example 7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5 e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( 0.5 0.4 • Convolution Theorem 0.3 0.2 1 |U(f)| • Convolution Example U(f) = [from before] 0.1 • a+i2πf -5 0 5 Convolution Properties Frequency (Hz) • Parseval’s Theorem 0.5 • Energy Conservation v(t) = cos 2πFt 0 • Energy Spectrum -0.5 0.4 F=2 0.5 0.5 W f U f V f 0.2 ( ) = ( ) ∗ ( ) |V(f)| 0 0.5 0.5 -5 0 5 = a+i2π(f+F ) + a+i2π(f−F ) Frequency (Hz) 1 a=2, F=2 0.5 w(t) 0 -0.5 -5 0 5 Time (s) 0.2 0.1 |W(f)| 0 -5 0 5 Frequency (Hz) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 Multiplication Example 7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5 e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( 0.5 0.4 • Convolution Theorem 0.3 0.2 1 |U(f)| • Convolution Example U(f) = [from before] 0.1 • a+i2πf -5 0 5 Convolution Properties Frequency (Hz) • Parseval’s Theorem 0.5 • Energy Conservation v(t) = cos 2πFt 0 • Energy Spectrum -0.5 0.4 F=2 0.5 0.5 W f U f V f 0.2 ( ) = ( ) ∗ ( ) |V(f)| 0 0.5 0.5 -5 0 5 = a+i2π(f+F ) + a+i2π(f−F ) Frequency (Hz) 1 a=2, F=2 0.5 w(t) 0 -0.5 -5 0 5 Time (s) 0.2 0.1 |W(f)| 0 -5 0 5 Frequency (Hz) 0.5 0 -0.5 E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 Multiplication Example 7: Fourier Transforms: 1 −at a=2 Convolution and Parseval’s 0.5 e t ≥ 0 u(t) Theorem u(t) = 0 • Multiplication of Signals -5 0 5 0 t < 0 Time (s) • Multiplication Example ( 0.5 0.4 • Convolution Theorem 0.3 0.2 1 |U(f)| • Convolution Example U(f) = [from before] 0.1 • a+i2πf -5 0 5 Convolution Properties Frequency (Hz) • Parseval’s Theorem 0.5 • Energy Conservation v(t) = cos 2πFt 0 • Energy Spectrum -0.5 0.4 F=2 0.5 0.5 W f U f V f 0.2 ( ) = ( ) ∗ ( ) |V(f)| 0 0.5 0.5 -5 0 5 = a+i2π(f+F ) + a+i2π(f−F ) Frequency (Hz) 1 a=2, F=2 0.5 w(t) 0 -0.5 If V (f) consists entirely of Dirac impulses -5 0 5 Time (s) then U(f) ∗ V (f) iust replaces each 0.2 0.1 |W(f)| impulse with a complete copy of U(f) 0 -5 0 5 Frequency (Hz) scaled by the area of the impulse and shifted 0.5 so that 0Hz lies on the impulse. Then add 0 -0.5 E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 3 / 10 Convolution Theorem 7: Fourier Transforms: Convolution and Parseval’s Convolution Theorem: Theorem w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Multiplication of Signals • Multiplication Example • Convolution Theorem • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 4 / 10 Convolution Theorem 7: Fourier Transforms: Convolution and Parseval’s Convolution Theorem: Theorem w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Multiplication of Signals • Multiplication Example w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) • Convolution Theorem • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 4 / 10 Convolution Theorem 7: Fourier Transforms: Convolution and Parseval’s Convolution Theorem: Theorem w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Multiplication of Signals • Multiplication Example w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) • Convolution Theorem • Convolution Example Convolution in the time domain is equivalent to multiplication in the • Convolution Properties • Parseval’s Theorem frequency domain and vice versa. • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 4 / 10 Convolution Theorem 7: Fourier Transforms: Convolution and Parseval’s Convolution Theorem: Theorem w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Multiplication of Signals • Multiplication Example w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) • Convolution Theorem • Convolution Example Convolution in the time domain is equivalent to multiplication in the • Convolution Properties • Parseval’s Theorem frequency domain and vice versa. • Energy Conservation • Energy Spectrum • Summary Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 4 / 10 Convolution Theorem 7: Fourier Transforms: Convolution and Parseval’s Convolution Theorem: Theorem w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Multiplication of Signals • Multiplication Example w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) • Convolution Theorem • Convolution Example Convolution in the time domain is equivalent to multiplication in the • Convolution Properties • Parseval’s Theorem frequency domain and vice versa. • Energy Conservation • Energy Spectrum • Summary Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) define dual waveforms x(t), y(t) and z(t) as follows: x(t) = U(t) ⇔ X(f) = u(−f) [duality] E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 4 / 10 Convolution Theorem 7: Fourier Transforms: Convolution and Parseval’s Convolution Theorem: Theorem w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Multiplication of Signals • Multiplication Example w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) • Convolution Theorem • Convolution Example Convolution in the time domain is equivalent to multiplication in the • Convolution Properties • Parseval’s Theorem frequency domain and vice versa. • Energy Conservation • Energy Spectrum • Summary Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) define dual waveforms x(t), y(t) and z(t) as follows: x(t) = U(t) ⇔ X(f) = u(−f) [duality] y(t) = V (t) ⇔ Y (f) = v(−f) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 4 / 10 Convolution Theorem 7: Fourier Transforms: Convolution and Parseval’s Convolution Theorem: Theorem w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Multiplication of Signals • Multiplication Example w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) • Convolution Theorem • Convolution Example Convolution in the time domain is equivalent to multiplication in the • Convolution Properties • Parseval’s Theorem frequency domain and vice versa. • Energy Conservation • Energy Spectrum • Summary Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) define dual waveforms x(t), y(t) and z(t) as follows: x(t) = U(t) ⇔ X(f) = u(−f) [duality] y(t) = V (t) ⇔ Y (f) = v(−f) z(t) = W (t) ⇔ Z(f) = w(−f) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 4 / 10 Convolution Theorem 7: Fourier Transforms: Convolution and Parseval’s Convolution Theorem: Theorem w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Multiplication of Signals • Multiplication Example w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) • Convolution Theorem • Convolution Example Convolution in the time domain is equivalent to multiplication in the • Convolution Properties • Parseval’s Theorem frequency domain and vice versa. • Energy Conservation • Energy Spectrum • Summary Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) define dual waveforms x(t), y(t) and z(t) as follows: x(t) = U(t) ⇔ X(f) = u(−f) [duality] y(t) = V (t) ⇔ Y (f) = v(−f) z(t) = W (t) ⇔ Z(f) = w(−f) Now the convolution property becomes: w(−f) = u(−f)v(−f) ⇔ W (t) = U(t) ∗ V (t) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 4 / 10 Convolution Theorem 7: Fourier Transforms: Convolution and Parseval’s Convolution Theorem: Theorem w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Multiplication of Signals • Multiplication Example w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) • Convolution Theorem • Convolution Example Convolution in the time domain is equivalent to multiplication in the • Convolution Properties • Parseval’s Theorem frequency domain and vice versa. • Energy Conservation • Energy Spectrum • Summary Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) define dual waveforms x(t), y(t) and z(t) as follows: x(t) = U(t) ⇔ X(f) = u(−f) [duality] y(t) = V (t) ⇔ Y (f) = v(−f) z(t) = W (t) ⇔ Z(f) = w(−f) Now the convolution property becomes: w(−f) = u(−f)v(−f) ⇔ W (t) = U(t) ∗ V (t) [sub t ↔ ±f] E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 4 / 10 Convolution Theorem 7: Fourier Transforms: Convolution and Parseval’s Convolution Theorem: Theorem w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Multiplication of Signals • Multiplication Example w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) • Convolution Theorem • Convolution Example Convolution in the time domain is equivalent to multiplication in the • Convolution Properties • Parseval’s Theorem frequency domain and vice versa. • Energy Conservation • Energy Spectrum • Summary Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) define dual waveforms x(t), y(t) and z(t) as follows: x(t) = U(t) ⇔ X(f) = u(−f) [duality] y(t) = V (t) ⇔ Y (f) = v(−f) z(t) = W (t) ⇔ Z(f) = w(−f) Now the convolution property becomes: w(−f) = u(−f)v(−f) ⇔ W (t) = U(t) ∗ V (t) [sub t ↔ ±f] Z(f) = X(f)Y (f) ⇔ z(t) = x(t) ∗ y(t) [duality] E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 4 / 10 ∫ ∫ ∫ Convolution Example 7: Fourier Transforms: Convolution and Parseval’s 1 − t 0 ≤ t < 1 Theorem u(t) = • Multiplication of Signals 0 otherwise • Multiplication Example ( • Convolution Theorem • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 ∫ ∫ ∫ Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 ∫ ∫ ∫ Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem • Convolution Example e−t t ≥ 0 • Convolution Properties v(t) = • Parseval’s Theorem 0 t < 0 • Energy Conservation ( • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 ∫ ∫ ∫ Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem 1 • Convolution Example −t 0.5 v(t) • e t ≥ 0 Convolution Properties v(t) = 0 • Parseval’s Theorem -2 0 2 4 6 0 t < 0 Time t (s) • Energy Conservation ( • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 ∫ ∫ ∫ Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem 1 • Convolution Example −t 0.5 v(t) • e t ≥ 0 Convolution Properties v(t) = 0 • Parseval’s Theorem -2 0 2 4 6 0 t < 0 Time t (s) • Energy Conservation ( • Energy Spectrum • Summary w(t) = u(t) ∗ v(t) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 ∫ ∫ ∫ Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem 1 • Convolution Example −t 0.5 v(t) • e t ≥ 0 Convolution Properties v(t) = 0 • Parseval’s Theorem -2 0 2 4 6 0 t < 0 Time t (s) • Energy Conservation ( • Energy Spectrum • Summary w(t) = u(t) ∗ v(t) ∞ = −∞ u(τ)v(t − τ)dτ R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 ∫ ∫ ∫ Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem 1 • Convolution Example −t 0.5 v(t) • e t ≥ 0 Convolution Properties v(t) = 0 • Parseval’s Theorem -2 0 2 4 6 0 t < 0 Time t (s) • Energy Conservation ( • Energy Spectrum • Summary w(t) = u(t) ∗ v(t) ∞ = −∞ u(τ)v(t − τ)dτ min(t,1) τ−t = R0 (1 − τ)e dτ R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 ∫ ∫ ∫ Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem 1 • Convolution Example −t 0.5 v(t) • e t ≥ 0 Convolution Properties v(t) = 0 • Parseval’s Theorem -2 0 2 4 6 0 t < 0 Time t (s) • Energy Conservation ( • Energy Spectrum • Summary w(t) = u(t) ∗ v(t) ∞ = −∞ u(τ)v(t − τ)dτ min(t,1) τ−t = R0 (1 − τ)e dτ τ−t min(t,1) = [(2R − τ) e ]τ=0 E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 ∫ ∫ ∫ Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem 1 • Convolution Example −t 0.5 v(t) • e t ≥ 0 Convolution Properties v(t) = 0 • Parseval’s Theorem -2 0 2 4 6 0 t < 0 Time t (s) • Energy Conservation ( • Energy Spectrum • Summary w(t) = u(t) ∗ v(t) ∞ = −∞ u(τ)v(t − τ)dτ min(t,1) τ−t = R0 (1 − τ)e dτ τ−t min(t,1) = [(2R − τ) e ]τ=0 0 t < 0 −t = 2 − t − 2e 0 ≤ t < 1 (e − 2) e−t t ≥ 1 E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 ∫ ∫ ∫ Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem 1 • Convolution Example −t 0.5 v(t) • e t ≥ 0 Convolution Properties v(t) = 0 • Parseval’s Theorem -2 0 2 4 6 0 t < 0 Time t (s) • Energy Conservation ( 0.3 • Energy Spectrum 0.2 • Summary w(t) 0.1 0 w(t) = u(t) ∗ v(t) -2 0 2 4 6 ∞ Time t (s) = −∞ u(τ)v(t − τ)dτ min(t,1) τ−t = R0 (1 − τ)e dτ τ−t min(t,1) = [(2R − τ) e ]τ=0 0 t < 0 −t = 2 − t − 2e 0 ≤ t < 1 (e − 2) e−t t ≥ 1 E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 ∫ ∫ Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem 1 • Convolution Example −t 0.5 v(t) • e t ≥ 0 Convolution Properties v(t) = 0 • Parseval’s Theorem -2 0 2 4 6 0 t < 0 Time t (s) • Energy Conservation ( 0.3 • Energy Spectrum 0.2 • Summary w(t) 0.1 0 w(t) = u(t) ∗ v(t) -2 0 2 4 6 ∞ Time t (s) 1 = u(τ)v(t − τ)dτ ∫ = 0.307 t=0.7 −∞ u(τ) v(0.7-τ) 0.5 min(t,1) τ−t 0 = R (1 − τ)e dτ -2 0 2 4 6 0 Time τ (s) τ−t min(t,1) = [(2R − τ) e ]τ=0 0 t < 0 −t = 2 − t − 2e 0 ≤ t < 1 (e − 2) e−t t ≥ 1 E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 ∫ Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem 1 • Convolution Example −t 0.5 v(t) • e t ≥ 0 Convolution Properties v(t) = 0 • Parseval’s Theorem -2 0 2 4 6 0 t < 0 Time t (s) • Energy Conservation ( 0.3 • Energy Spectrum 0.2 • Summary w(t) 0.1 0 w(t) = u(t) ∗ v(t) -2 0 2 4 6 ∞ Time t (s) 1 = u(τ)v(t − τ)dτ ∫ = 0.307 t=0.7 −∞ u(τ) v(0.7-τ) 0.5 min(t,1) τ−t 0 = R (1 − τ)e dτ -2 0 2 4 6 0 Time τ (s) 1 τ−t min(t,1) ∫ = 0.16 t=1.5 u(τ) v(1.5-τ) = [(2R − τ) e ]τ=0 0.5 0 -2 0 2 4 6 0 t < 0 Time τ (s) −t = 2 − t − 2e 0 ≤ t < 1 (e − 2) e−t t ≥ 1 E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem 1 • Convolution Example −t 0.5 v(t) • e t ≥ 0 Convolution Properties v(t) = 0 • Parseval’s Theorem -2 0 2 4 6 0 t < 0 Time t (s) • Energy Conservation ( 0.3 • Energy Spectrum 0.2 • Summary w(t) 0.1 0 w(t) = u(t) ∗ v(t) -2 0 2 4 6 ∞ Time t (s) 1 = u(τ)v(t − τ)dτ ∫ = 0.307 t=0.7 −∞ u(τ) v(0.7-τ) 0.5 min(t,1) τ−t 0 = R (1 − τ)e dτ -2 0 2 4 6 0 Time τ (s) 1 τ−t min(t,1) ∫ = 0.16 t=1.5 u(τ) v(1.5-τ) = [(2R − τ) e ]τ=0 0.5 0 -2 0 2 4 6 0 t < 0 Time τ (s) 1 −t ∫ = 0.059 t=2.5 = 2 − t − 2e 0 ≤ t < 1 u(τ) v(2.5-τ) 0.5 −t 0 (e − 2) e t ≥ 1 -2 0 2 4 6 Time τ (s) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 Convolution Example 7: Fourier Transforms: 1 Convolution and Parseval’s 1 − t 0 ≤ t < 1 0.5 Theorem u t u(t) • ( ) = 0 Multiplication of Signals 0 otherwise -2 0 2 4 6 • Multiplication Example ( Time t (s) • Convolution Theorem 1 • Convolution Example −t 0.5 v(t) • e t ≥ 0 Convolution Properties v(t) = 0 • Parseval’s Theorem -2 0 2 4 6 0 t < 0 Time t (s) • Energy Conservation ( 0.3 • Energy Spectrum 0.2 • Summary w(t) 0.1 0 w(t) = u(t) ∗ v(t) -2 0 2 4 6 ∞ Time t (s) 1 = u(τ)v(t − τ)dτ ∫ = 0.307 t=0.7 −∞ u(τ) v(0.7-τ) 0.5 min(t,1) τ−t 0 = R (1 − τ)e dτ -2 0 2 4 6 0 Time τ (s) 1 τ−t min(t,1) ∫ = 0.16 t=1.5 u(τ) v(1.5-τ) = [(2R − τ) e ]τ=0 0.5 0 -2 0 2 4 6 0 t < 0 Time τ (s) 1 −t ∫ = 0.059 t=2.5 = 2 − t − 2e 0 ≤ t < 1 u(τ) v(2.5-τ) 0.5 −t 0 (e − 2) e t ≥ 1 -2 0 2 4 6 Time τ (s) Note howv(t − τ) is time-reversed (because of the −τ ) and time-shifted to put the time origin at τ = t. E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 5 / 10 Convolution Properties 7: Fourier Transforms: ∞ Convolution and Parseval’s Convloution: w(t) = u(t) ∗ v(t) , u(τ)v(t − τ)dτ Theorem −∞ • Multiplication of Signals • Multiplication Example R • Convolution Theorem • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 6 / 10 Convolution Properties 7: Fourier Transforms: ∞ Convolution and Parseval’s Convloution: w(t) = u(t) ∗ v(t) , u(τ)v(t − τ)dτ Theorem −∞ • Multiplication of Signals • Multiplication Example Convolution behaves algebraically likeR multiplication: • Convolution Theorem • Convolution Example 1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t) • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 6 / 10 Convolution Properties 7: Fourier Transforms: ∞ Convolution and Parseval’s Convloution: w(t) = u(t) ∗ v(t) , u(τ)v(t − τ)dτ Theorem −∞ • Multiplication of Signals • Multiplication Example Convolution behaves algebraically likeR multiplication: • Convolution Theorem • Convolution Example 1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t) • Convolution Properties • Parseval’s Theorem 2) Associative: • Energy Conservation u(t) ∗ v(t) ∗ w(t)=(u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t)) • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 6 / 10 Convolution Properties 7: Fourier Transforms: ∞ Convolution and Parseval’s Convloution: w(t) = u(t) ∗ v(t) , u(τ)v(t − τ)dτ Theorem −∞ • Multiplication of Signals • Multiplication Example Convolution behaves algebraically likeR multiplication: • Convolution Theorem • Convolution Example 1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t) • Convolution Properties • Parseval’s Theorem 2) Associative: • Energy Conservation u(t) ∗ v(t) ∗ w(t)=(u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t)) • Energy Spectrum • Summary 3) Distributive over addition: w(t) ∗ (u(t) + v(t)) = w(t) ∗ u(t) + w(t) ∗ v(t) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 6 / 10 Convolution Properties 7: Fourier Transforms: ∞ Convolution and Parseval’s Convloution: w(t) = u(t) ∗ v(t) , u(τ)v(t − τ)dτ Theorem −∞ • Multiplication of Signals • Multiplication Example Convolution behaves algebraically likeR multiplication: • Convolution Theorem • Convolution Example 1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t) • Convolution Properties • Parseval’s Theorem 2) Associative: • Energy Conservation u(t) ∗ v(t) ∗ w(t)=(u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t)) • Energy Spectrum • Summary 3) Distributive over addition: w(t) ∗ (u(t) + v(t)) = w(t) ∗ u(t) + w(t) ∗ v(t) 4) Identity Element or “1”: u(t) ∗ δ(t) = δ(t) ∗ u(t) = u(t) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 6 / 10 Convolution Properties 7: Fourier Transforms: ∞ Convolution and Parseval’s Convloution: w(t) = u(t) ∗ v(t) , u(τ)v(t − τ)dτ Theorem −∞ • Multiplication of Signals • Multiplication Example Convolution behaves algebraically likeR multiplication: • Convolution Theorem • Convolution Example 1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t) • Convolution Properties • Parseval’s Theorem 2) Associative: • Energy Conservation u(t) ∗ v(t) ∗ w(t)=(u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t)) • Energy Spectrum • Summary 3) Distributive over addition: w(t) ∗ (u(t) + v(t)) = w(t) ∗ u(t) + w(t) ∗ v(t) 4) Identity Element or “1”: u(t) ∗ δ(t) = δ(t) ∗ u(t) = u(t) 5) Bilinear: (au(t)) ∗ (bv(t)) = ab (u(t) ∗ v(t)) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 6 / 10 Convolution Properties 7: Fourier Transforms: ∞ Convolution and Parseval’s Convloution: w(t) = u(t) ∗ v(t) , u(τ)v(t − τ)dτ Theorem −∞ • Multiplication of Signals • Multiplication Example Convolution behaves algebraically likeR multiplication: • Convolution Theorem • Convolution Example 1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t) • Convolution Properties • Parseval’s Theorem 2) Associative: • Energy Conservation u(t) ∗ v(t) ∗ w(t)=(u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t)) • Energy Spectrum • Summary 3) Distributive over addition: w(t) ∗ (u(t) + v(t)) = w(t) ∗ u(t) + w(t) ∗ v(t) 4) Identity Element or “1”: u(t) ∗ δ(t) = δ(t) ∗ u(t) = u(t) 5) Bilinear: (au(t)) ∗ (bv(t)) = ab (u(t) ∗ v(t)) Proof: In the frequency domain, convolution is multiplication. E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 6 / 10 Convolution Properties 7: Fourier Transforms: ∞ Convolution and Parseval’s Convloution: w(t) = u(t) ∗ v(t) , u(τ)v(t − τ)dτ Theorem −∞ • Multiplication of Signals • Multiplication Example Convolution behaves algebraically likeR multiplication: • Convolution Theorem • Convolution Example 1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t) • Convolution Properties • Parseval’s Theorem 2) Associative: • Energy Conservation u(t) ∗ v(t) ∗ w(t)=(u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t)) • Energy Spectrum • Summary 3) Distributive over addition: w(t) ∗ (u(t) + v(t)) = w(t) ∗ u(t) + w(t) ∗ v(t) 4) Identity Element or “1”: u(t) ∗ δ(t) = δ(t) ∗ u(t) = u(t) 5) Bilinear: (au(t)) ∗ (bv(t)) = ab (u(t) ∗ v(t)) Proof: In the frequency domain, convolution is multiplication. Also, if u(t) ∗ v(t) = w(t), then 6) Time Shifting: u(t + a) ∗ v(t + b) = w(t + a + b) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 6 / 10 Convolution Properties 7: Fourier Transforms: ∞ Convolution and Parseval’s Convloution: w(t) = u(t) ∗ v(t) , u(τ)v(t − τ)dτ Theorem −∞ • Multiplication of Signals • Multiplication Example Convolution behaves algebraically likeR multiplication: • Convolution Theorem • Convolution Example 1) Commutative: u(t) ∗ v(t) = v(t) ∗ u(t) • Convolution Properties • Parseval’s Theorem 2) Associative: • Energy Conservation u(t) ∗ v(t) ∗ w(t)=(u(t) ∗ v(t)) ∗ w(t) = u(t) ∗ (v(t) ∗ w(t)) • Energy Spectrum • Summary 3) Distributive over addition: w(t) ∗ (u(t) + v(t)) = w(t) ∗ u(t) + w(t) ∗ v(t) 4) Identity Element or “1”: u(t) ∗ δ(t) = δ(t) ∗ u(t) = u(t) 5) Bilinear: (au(t)) ∗ (bv(t)) = ab (u(t) ∗ v(t)) Proof: In the frequency domain, convolution is multiplication. Also, if u(t) ∗ v(t) = w(t), then 6) Time Shifting: u(t + a) ∗ v(t + b) = w(t + a + b) 1 7) Time Scaling: u(at) ∗ v(at) = |a| w(at) How to recognise a convolution integral: the arguments of u(··· ) and v(··· ) sum to a constant. E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 6 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem • Multiplication of Signals X(f) = δ(f − g) • Multiplication Example • Convolution Theorem • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df • Multiplication Example • Convolution Theorem R • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft i2πgt • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df = e • Multiplication Example • Convolution Theorem R • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft i2πgt • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df = e • Multiplication Example i2πgt −i2πft • Convolution Theorem ⇒ X(f) = e e R dt • Convolution Example • Convolution Properties R • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft i2πgt • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df = e • Multiplication Example i2πgt −i2πft i2π(g−f)t • Convolution Theorem ⇒ X(f) = e e R dt = e dt • Convolution Example • Convolution Properties R R • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft i2πgt • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df = e • Multiplication Example i2πgt −i2πft i2π(g−f)t • Convolution Theorem ⇒ X(f) = e e R dt = e dt = δ(g − f) • Convolution Example • Convolution Properties R R • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft i2πgt • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df = e • Multiplication Example i2πgt −i2πft i2π(g−f)t • Convolution Theorem ⇒ X(f) = e e R dt = e dt = δ(g − f) • Convolution Example • Convolution Properties ∞ R +R∞ • Parseval’s Theorem Parseval’s Theorem: u∗(t)v(t)dt = U ∗(f)V (f)df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary R R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft i2πgt • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df = e • Multiplication Example i2πgt −i2πft i2π(g−f)t • Convolution Theorem ⇒ X(f) = e e R dt = e dt = δ(g − f) • Convolution Example • Convolution Properties ∞ R +R∞ • Parseval’s Theorem Parseval’s Theorem: u∗(t)v(t)dt = U ∗(f)V (f)df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Proof: R R +∞ +∞ Let i2πft and i2πgt u(t) = f=−∞ U(f)e df v(t) = g=−∞ V (g)e dg R R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft i2πgt • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df = e • Multiplication Example i2πgt −i2πft i2π(g−f)t • Convolution Theorem ⇒ X(f) = e e R dt = e dt = δ(g − f) • Convolution Example • Convolution Properties ∞ R +R∞ • Parseval’s Theorem Parseval’s Theorem: u∗(t)v(t)dt = U ∗(f)V (f)df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Proof: R R +∞ +∞ Let i2πft and i2πgt u(t) = f=−∞ U(f)e df v(t) = g=−∞ V (g)e dg R R Now multiply u∗(t) = u(t) and v(t) together and integrate over time: E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft i2πgt • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df = e • Multiplication Example i2πgt −i2πft i2π(g−f)t • Convolution Theorem ⇒ X(f) = e e R dt = e dt = δ(g − f) • Convolution Example • Convolution Properties ∞ R +R∞ • Parseval’s Theorem Parseval’s Theorem: u∗(t)v(t)dt = U ∗(f)V (f)df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Proof: R R +∞ +∞ Let i2πft and i2πgt u(t) = f=−∞ U(f)e df v(t) = g=−∞ V (g)e dg R R Now multiply u∗(t) = u(t) and v(t) together and integrate over time: ∞ ∗ t=−∞ u (t)v(t)dt R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft i2πgt • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df = e • Multiplication Example i2πgt −i2πft i2π(g−f)t • Convolution Theorem ⇒ X(f) = e e R dt = e dt = δ(g − f) • Convolution Example • Convolution Properties ∞ R +R∞ • Parseval’s Theorem Parseval’s Theorem: u∗(t)v(t)dt = U ∗(f)V (f)df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Proof: R R +∞ +∞ Let i2πft and i2πgt u(t) = f=−∞ U(f)e df v(t) = g=−∞ V (g)e dg R [Note use of differentR dummy variables] Now multiply u∗(t) = u(t) and v(t) together and integrate over time: ∞ ∗ t=−∞ u (t)v(t)dt ∞ +∞ U ∗ f e−i2πftdf +∞ V g ei2πgtdgdt R = t=−∞ f=−∞ ( ) g=−∞ ( ) R R R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft i2πgt • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df = e • Multiplication Example i2πgt −i2πft i2π(g−f)t • Convolution Theorem ⇒ X(f) = e e R dt = e dt = δ(g − f) • Convolution Example • Convolution Properties ∞ R +R∞ • Parseval’s Theorem Parseval’s Theorem: u∗(t)v(t)dt = U ∗(f)V (f)df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Proof: R R +∞ +∞ Let i2πft and i2πgt u(t) = f=−∞ U(f)e df v(t) = g=−∞ V (g)e dg R [Note use of differentR dummy variables] Now multiply u∗(t) = u(t) and v(t) together and integrate over time: ∞ ∗ t=−∞ u (t)v(t)dt ∞ +∞ U ∗ f e−i2πftdf +∞ V g ei2πgtdgdt R = t=−∞ f=−∞ ( ) g=−∞ ( ) +∞ ∗ +∞ ∞ i2π(g−f)t = Rf=−∞RU (f) g=−∞ V (g) t=−∞R e dtdgdf R R R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft i2πgt • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df = e • Multiplication Example i2πgt −i2πft i2π(g−f)t • Convolution Theorem ⇒ X(f) = e e R dt = e dt = δ(g − f) • Convolution Example • Convolution Properties ∞ R +R∞ • Parseval’s Theorem Parseval’s Theorem: u∗(t)v(t)dt = U ∗(f)V (f)df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Proof: R R +∞ +∞ Let i2πft and i2πgt u(t) = f=−∞ U(f)e df v(t) = g=−∞ V (g)e dg R [Note use of differentR dummy variables] Now multiply u∗(t) = u(t) and v(t) together and integrate over time: ∞ ∗ t=−∞ u (t)v(t)dt ∞ +∞ U ∗ f e−i2πftdf +∞ V g ei2πgtdgdt R = t=−∞ f=−∞ ( ) g=−∞ ( ) +∞ ∗ +∞ ∞ i2π(g−f)t = Rf=−∞RU (f) g=−∞ V (g) t=−∞R e dtdgdf +∞ +∞ ∗ [lemma] = Rf=−∞ U (f) Rg=−∞ V (g)δR(g − f)dgdf R R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 7 / 10 Parseval's Theorem 7: Fourier Transforms: Convolution and Parseval’s Lemma: Theorem i2πft i2πgt • Multiplication of Signals X(f) = δ(f − g) ⇒ x(t) = δ(f − g)e df = e • Multiplication Example i2πgt −i2πft i2π(g−f)t • Convolution Theorem ⇒ X(f) = e e R dt = e dt = δ(g − f) • Convolution Example • Convolution Properties ∞ R +R∞ • Parseval’s Theorem Parseval’s Theorem: u∗(t)v(t)dt = U ∗(f)V (f)df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Proof: R R +∞ +∞ Let i2πft and i2πgt u(t) = f=−∞ U(f)e df v(t) = g=−∞ V (g)e dg R [Note use of differentR dummy variables] Now multiply u∗(t) = u(t) and v(t) together and integrate over time: ∞ ∗ t=−∞ u (t)v(t)dt ∞ +∞ U ∗ f e−i2πftdf +∞ V g ei2πgtdgdt R = t=−∞ f=−∞ ( ) g=−∞ ( ) +∞ ∗ +∞ ∞ i2π(g−f)t = Rf=−∞RU (f) g=−∞ V (g) t=−∞R e dtdgdf +∞ +∞ ∗ [lemma] = Rf=−∞ U (f) Rg=−∞ V (g)δR(g − f)dgdf +∞ ∗ = Rf=−∞ U (f)VR (f)df E1.10FourierSeriesandTransforms(2014-5559)R FourierTransform - Parseval and Convolution: 7 – 7 / 10 Energy Conservation 7: Fourier Transforms: ∞ ∗ +∞ ∗ Convolution and Parseval’s Parseval’s Theorem: u (t)v(t)dt = U (f)V (f)df Theorem t=−∞ f=−∞ • Multiplication of Signals • Multiplication Example R R • Convolution Theorem • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 8 / 10 Energy Conservation 7: Fourier Transforms: ∞ ∗ +∞ ∗ Convolution and Parseval’s Parseval’s Theorem: u (t)v(t)dt = U (f)V (f)df Theorem t=−∞ f=−∞ • Multiplication of Signals For the special case v(t) = u(t), Parseval’s theorem becomes: • Multiplication Example R R • Convolution Theorem ∞ ∗ +∞ ∗ • Convolution Example t=−∞ u (t)u(t)dt = f=−∞ U (f)U(f)df • Convolution Properties • Parseval’s Theorem R R • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 8 / 10 Energy Conservation 7: Fourier Transforms: ∞ ∗ +∞ ∗ Convolution and Parseval’s Parseval’s Theorem: u (t)v(t)dt = U (f)V (f)df Theorem t=−∞ f=−∞ • Multiplication of Signals For the special case v(t) = u(t), Parseval’s theorem becomes: • Multiplication Example R R • Convolution Theorem ∞ ∗ +∞ ∗ • Convolution Example t=−∞ u (t)u(t)dt = f=−∞ U (f)U(f)df • Convolution Properties • ∞ 2 +∞ 2 Parseval’s Theorem ⇒R Eu = |u(t)R| dt = |U(f)| df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary R R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 8 / 10 Energy Conservation 7: Fourier Transforms: ∞ ∗ +∞ ∗ Convolution and Parseval’s Parseval’s Theorem: u (t)v(t)dt = U (f)V (f)df Theorem t=−∞ f=−∞ • Multiplication of Signals For the special case v(t) = u(t), Parseval’s theorem becomes: • Multiplication Example R R • Convolution Theorem ∞ ∗ +∞ ∗ • Convolution Example t=−∞ u (t)u(t)dt = f=−∞ U (f)U(f)df • Convolution Properties • ∞ 2 +∞ 2 Parseval’s Theorem ⇒R Eu = |u(t)R| dt = |U(f)| df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Energy Conservation: The energyR in u(t) equals the energyR in U(f). E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 8 / 10 Energy Conservation 7: Fourier Transforms: ∞ ∗ +∞ ∗ Convolution and Parseval’s Parseval’s Theorem: u (t)v(t)dt = U (f)V (f)df Theorem t=−∞ f=−∞ • Multiplication of Signals For the special case v(t) = u(t), Parseval’s theorem becomes: • Multiplication Example R R • Convolution Theorem ∞ ∗ +∞ ∗ • Convolution Example t=−∞ u (t)u(t)dt = f=−∞ U (f)U(f)df • Convolution Properties • ∞ 2 +∞ 2 Parseval’s Theorem ⇒R Eu = |u(t)R| dt = |U(f)| df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Energy Conservation: The energyR in u(t) equals the energyR in U(f). Example: e−at t ≥ 0 u(t) = (0 t < 0 1 a=2 0.5 u(t) 0 -5 0 5 Time (s) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 8 / 10 Energy Conservation 7: Fourier Transforms: ∞ ∗ +∞ ∗ Convolution and Parseval’s Parseval’s Theorem: u (t)v(t)dt = U (f)V (f)df Theorem t=−∞ f=−∞ • Multiplication of Signals For the special case v(t) = u(t), Parseval’s theorem becomes: • Multiplication Example R R • Convolution Theorem ∞ ∗ +∞ ∗ • Convolution Example t=−∞ u (t)u(t)dt = f=−∞ U (f)U(f)df • Convolution Properties • ∞ 2 +∞ 2 Parseval’s Theorem ⇒R Eu = |u(t)R| dt = |U(f)| df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Energy Conservation: The energyR in u(t) equals the energyR in U(f). Example: −at −2 ∞ e t ≥ 0 2 −e at u(t) = ⇒ Eu = |u(t)| dt = 2a (0 t < 0 0 R h i 1 a=2 0.5 u(t) 0 -5 0 5 Time (s) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 8 / 10 Energy Conservation 7: Fourier Transforms: ∞ ∗ +∞ ∗ Convolution and Parseval’s Parseval’s Theorem: u (t)v(t)dt = U (f)V (f)df Theorem t=−∞ f=−∞ • Multiplication of Signals For the special case v(t) = u(t), Parseval’s theorem becomes: • Multiplication Example R R • Convolution Theorem ∞ ∗ +∞ ∗ • Convolution Example t=−∞ u (t)u(t)dt = f=−∞ U (f)U(f)df • Convolution Properties • ∞ 2 +∞ 2 Parseval’s Theorem ⇒R Eu = |u(t)R| dt = |U(f)| df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Energy Conservation: The energyR in u(t) equals the energyR in U(f). Example: −at −2 ∞ e t ≥ 0 2 −e at 1 u(t) = ⇒ Eu = |u(t)| dt = 2a = 2a (0 t < 0 0 R h i 1 a=2 0.5 u(t) 0 -5 0 5 Time (s) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 8 / 10 Energy Conservation 7: Fourier Transforms: ∞ ∗ +∞ ∗ Convolution and Parseval’s Parseval’s Theorem: u (t)v(t)dt = U (f)V (f)df Theorem t=−∞ f=−∞ • Multiplication of Signals For the special case v(t) = u(t), Parseval’s theorem becomes: • Multiplication Example R R • Convolution Theorem ∞ ∗ +∞ ∗ • Convolution Example t=−∞ u (t)u(t)dt = f=−∞ U (f)U(f)df • Convolution Properties • ∞ 2 +∞ 2 Parseval’s Theorem ⇒R Eu = |u(t)R| dt = |U(f)| df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Energy Conservation: The energyR in u(t) equals the energyR in U(f). Example: −at −2 ∞ e t ≥ 0 2 −e at 1 u(t) = ⇒ Eu = |u(t)| dt = 2a = 2a (0 t < 0 0 R h i 1 1 a=2 U(f) = [from before] 0.5 a+i2πf u(t) 0 -5 0 5 Time (s) 0.5 0.4 0.3 0.2 |U(f)| 0.1 -5 0 5 Frequency (Hz) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 8 / 10 Energy Conservation 7: Fourier Transforms: ∞ ∗ +∞ ∗ Convolution and Parseval’s Parseval’s Theorem: u (t)v(t)dt = U (f)V (f)df Theorem t=−∞ f=−∞ • Multiplication of Signals For the special case v(t) = u(t), Parseval’s theorem becomes: • Multiplication Example R R • Convolution Theorem ∞ ∗ +∞ ∗ • Convolution Example t=−∞ u (t)u(t)dt = f=−∞ U (f)U(f)df • Convolution Properties • ∞ 2 +∞ 2 Parseval’s Theorem ⇒R Eu = |u(t)R| dt = |U(f)| df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Energy Conservation: The energyR in u(t) equals the energyR in U(f). Example: −at −2 ∞ e t ≥ 0 2 −e at 1 u(t) = ⇒ Eu = |u(t)| dt = 2a = 2a (0 t < 0 0 R h i 1 1 a=2 U(f) = [from before] 0.5 a+i2πf u(t) 0 2 df -5 0 5 ⇒ |U(f)| df = 2 2 2 Time (s) a +4π f 0.5 0.4 0.3 0.2 |U(f)| 0.1 R R -5 0 5 Frequency (Hz) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 8 / 10 Energy Conservation 7: Fourier Transforms: ∞ ∗ +∞ ∗ Convolution and Parseval’s Parseval’s Theorem: u (t)v(t)dt = U (f)V (f)df Theorem t=−∞ f=−∞ • Multiplication of Signals For the special case v(t) = u(t), Parseval’s theorem becomes: • Multiplication Example R R • Convolution Theorem ∞ ∗ +∞ ∗ • Convolution Example t=−∞ u (t)u(t)dt = f=−∞ U (f)U(f)df • Convolution Properties • ∞ 2 +∞ 2 Parseval’s Theorem ⇒R Eu = |u(t)R| dt = |U(f)| df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Energy Conservation: The energyR in u(t) equals the energyR in U(f). Example: −at −2 ∞ e t ≥ 0 2 −e at 1 u(t) = ⇒ Eu = |u(t)| dt = 2a = 2a (0 t < 0 0 R h i 1 1 a=2 U(f) = [from before] 0.5 a+i2πf u(t) 0 2 df -5 0 5 ⇒ |U(f)| df = 2 2 2 Time (s) a +4π f 0.5 0.4 0.3 −1 2πf ∞ 0.2 |U(f)| 0.1 R tan ( a ) R = -5 0 5 2πa Frequency (Hz) −∞ E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 8 / 10 Energy Conservation 7: Fourier Transforms: ∞ ∗ +∞ ∗ Convolution and Parseval’s Parseval’s Theorem: u (t)v(t)dt = U (f)V (f)df Theorem t=−∞ f=−∞ • Multiplication of Signals For the special case v(t) = u(t), Parseval’s theorem becomes: • Multiplication Example R R • Convolution Theorem ∞ ∗ +∞ ∗ • Convolution Example t=−∞ u (t)u(t)dt = f=−∞ U (f)U(f)df • Convolution Properties • ∞ 2 +∞ 2 Parseval’s Theorem ⇒R Eu = |u(t)R| dt = |U(f)| df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Energy Conservation: The energyR in u(t) equals the energyR in U(f). Example: −at −2 ∞ e t ≥ 0 2 −e at 1 u(t) = ⇒ Eu = |u(t)| dt = 2a = 2a (0 t < 0 0 R h i 1 1 a=2 U(f) = [from before] 0.5 a+i2πf u(t) 0 2 df -5 0 5 ⇒ |U(f)| df = 2 2 2 Time (s) a +4π f 0.5 0.4 0.3 −1 2πf ∞ 0.2 |U(f)| 0.1 R tan ( a ) R π = = -5 0 5 2πa 2πa Frequency (Hz) −∞ E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 8 / 10 Energy Conservation 7: Fourier Transforms: ∞ ∗ +∞ ∗ Convolution and Parseval’s Parseval’s Theorem: u (t)v(t)dt = U (f)V (f)df Theorem t=−∞ f=−∞ • Multiplication of Signals For the special case v(t) = u(t), Parseval’s theorem becomes: • Multiplication Example R R • Convolution Theorem ∞ ∗ +∞ ∗ • Convolution Example t=−∞ u (t)u(t)dt = f=−∞ U (f)U(f)df • Convolution Properties • ∞ 2 +∞ 2 Parseval’s Theorem ⇒R Eu = |u(t)R| dt = |U(f)| df • Energy Conservation t=−∞ f=−∞ • Energy Spectrum • Summary Energy Conservation: The energyR in u(t) equals the energyR in U(f). Example: −at −2 ∞ e t ≥ 0 2 −e at 1 u(t) = ⇒ Eu = |u(t)| dt = 2a = 2a (0 t < 0 0 R h i 1 1 a=2 U(f) = [from before] 0.5 a+i2πf u(t) 0 2 df -5 0 5 ⇒ |U(f)| df = 2 2 2 Time (s) a +4π f 0.5 0.4 0.3 −1 2πf ∞ 0.2 |U(f)| 0.1 R tan ( a ) R π 1 = = = -5 0 5 2πa 2πa 2a Frequency (Hz) −∞ E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 8 / 10 < Energy Spectrum 7: Fourier Transforms: Convolution and Parseval’s Example from before: Theorem −at • Multiplication of Signals e cos 2πFt t ≥ 0 • Multiplication Example w(t) = • Convolution Theorem (0 t < 0 • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 9 / 10 < Energy Spectrum 7: Fourier Transforms: 1 a=2, F=2 Convolution and Parseval’s Example from before: 0.5 w(t) 0 Theorem −at -0.5 • Multiplication of Signals -5 0 5 e cos 2πFt t ≥ 0 Time (s) • Multiplication Example w(t) = • Convolution Theorem (0 t < 0 • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 9 / 10 < Energy Spectrum 7: Fourier Transforms: 1 a=2, F=2 Convolution and Parseval’s Example from before: 0.5 w(t) 0 Theorem −at -0.5 • Multiplication of Signals -5 0 5 e cos 2πFt t ≥ 0 Time (s) • Multiplication Example w(t) = • Convolution Theorem (0 t < 0 • Convolution Example • Convolution Properties 0.5 0.5 • Parseval’s Theorem W (f) = a+i2π(f+F ) + a+i2π(f−F ) • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 9 / 10 < Energy Spectrum 7: Fourier Transforms: 1 a=2, F=2 Convolution and Parseval’s Example from before: 0.5 w(t) 0 Theorem −at -0.5 • Multiplication of Signals -5 0 5 e cos 2πFt t ≥ 0 Time (s) • Multiplication Example w(t) = • Convolution Theorem (0 t < 0 • Convolution Example • Convolution Properties 0.5 0.5 • Parseval’s Theorem W (f) = a+i2π(f+F ) + a+i2π(f−F ) • Energy Conservation • Energy Spectrum a+i2πf • Summary = a2+i4πaf−4π2(f 2−F 2) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 9 / 10 Energy Spectrum 7: Fourier Transforms: 1 a=2, F=2 Convolution and Parseval’s Example from before: 0.5 w(t) 0 Theorem −at -0.5 • Multiplication of Signals -5 0 5 e cos 2πFt t ≥ 0 Time (s) • Multiplication Example w(t) = 0.25 • 0.2 Convolution Theorem 0 t < 0 0.15 ( 0.1 • Convolution Example |W(f)| 0.05 • -5 0 5 Convolution Properties 0.5 0.5 Frequency (Hz) • Parseval’s Theorem W (f) = a+i2π(f+F ) + a+i2π(f−F ) 0.5 • Energy Conservation 0 • Energy Spectrum a+i2πf -0.5 2 2 2 2 (rad/pi) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 9 / 10 Energy Spectrum 7: Fourier Transforms: 1 a=2, F=2 Convolution and Parseval’s Example from before: 0.5 w(t) 0 Theorem −at -0.5 • Multiplication of Signals -5 0 5 e cos 2πFt t ≥ 0 Time (s) • Multiplication Example w(t) = 0.25 • 0.2 Convolution Theorem 0 t < 0 0.15 ( 0.1 • Convolution Example |W(f)| 0.05 • -5 0 5 Convolution Properties 0.5 0.5 Frequency (Hz) • Parseval’s Theorem W (f) = a+i2π(f+F ) + a+i2π(f−F ) 0.5 • Energy Conservation 0 • Energy Spectrum a+i2πf -0.5 2 2 2 2 (rad/pi) 2 2 2 2 a +4π f W f 2 | ( )| = (a2−4π2(f 2−F 2)) +16π2a2f 2 E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 9 / 10 Energy Spectrum 7: Fourier Transforms: 1 a=2, F=2 Convolution and Parseval’s Example from before: 0.5 w(t) 0 Theorem −at -0.5 • Multiplication of Signals -5 0 5 e cos 2πFt t ≥ 0 Time (s) • Multiplication Example w(t) = 0.25 • 0.2 Convolution Theorem 0 t < 0 0.15 ( 0.1 • Convolution Example |W(f)| 0.05 • -5 0 5 Convolution Properties 0.5 0.5 Frequency (Hz) • Parseval’s Theorem W (f) = a+i2π(f+F ) + a+i2π(f−F ) 0.5 • Energy Conservation 0 • Energy Spectrum a+i2πf -0.5 2 2 2 2 (rad/pi) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 9 / 10 Energy Spectrum 7: Fourier Transforms: 1 a=2, F=2 Convolution and Parseval’s Example from before: 0.5 w(t) 0 Theorem −at -0.5 • Multiplication of Signals -5 0 5 e cos 2πFt t ≥ 0 Time (s) • Multiplication Example w(t) = 0.25 • 0.2 Convolution Theorem 0 t < 0 0.15 ( 0.1 • Convolution Example |W(f)| 0.05 • -5 0 5 Convolution Properties 0.5 0.5 Frequency (Hz) • Parseval’s Theorem W (f) = a+i2π(f+F ) + a+i2π(f−F ) 0.5 • Energy Conservation 0 • Energy Spectrum a+i2πf -0.5 2 2 2 2 (rad/pi) • The units of |W (f)|2 are “energy per Hz” so that its integral, ∞ 2 , has units of energy. Ew = −∞ |W (f)| df R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 9 / 10 Energy Spectrum 7: Fourier Transforms: 1 a=2, F=2 Convolution and Parseval’s Example from before: 0.5 w(t) 0 Theorem −at -0.5 • Multiplication of Signals -5 0 5 e cos 2πFt t ≥ 0 Time (s) • Multiplication Example w(t) = 0.25 • 0.2 Convolution Theorem 0 t < 0 0.15 ( 0.1 • Convolution Example |W(f)| 0.05 • -5 0 5 Convolution Properties 0.5 0.5 Frequency (Hz) • Parseval’s Theorem W (f) = a+i2π(f+F ) + a+i2π(f−F ) 0.5 • Energy Conservation 0 • Energy Spectrum a+i2πf -0.5 2 2 2 2 (rad/pi) Energy Spectrum • The units of |W (f)|2 are “energy per Hz” so that its integral, ∞ 2 , has units of energy. Ew = −∞ |W (f)| df 2 • The quantityR |W (f)| is called the energy spectral density of w(t) at frequency f and its graph is the energy spectrum of w(t). E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 9 / 10 Energy Spectrum 7: Fourier Transforms: 1 a=2, F=2 Convolution and Parseval’s Example from before: 0.5 w(t) 0 Theorem −at -0.5 • Multiplication of Signals -5 0 5 e cos 2πFt t ≥ 0 Time (s) • Multiplication Example w(t) = 0.25 • 0.2 Convolution Theorem 0 t < 0 0.15 ( 0.1 • Convolution Example |W(f)| 0.05 • -5 0 5 Convolution Properties 0.5 0.5 Frequency (Hz) • Parseval’s Theorem W (f) = a+i2π(f+F ) + a+i2π(f−F ) 0.5 • Energy Conservation 0 • Energy Spectrum a+i2πf -0.5 2 2 2 2 (rad/pi) Energy Spectrum • The units of |W (f)|2 are “energy per Hz” so that its integral, ∞ 2 , has units of energy. Ew = −∞ |W (f)| df 2 • The quantityR |W (f)| is called the energy spectral density of w(t) at frequency f and its graph is the energy spectrum of w(t). It shows how the energy of w(t) is distributed over frequencies. E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 9 / 10 Energy Spectrum 7: Fourier Transforms: 1 a=2, F=2 Convolution and Parseval’s Example from before: 0.5 w(t) 0 Theorem −at -0.5 • Multiplication of Signals -5 0 5 e cos 2πFt t ≥ 0 Time (s) • Multiplication Example w(t) = 0.25 • 0.2 Convolution Theorem 0 t < 0 0.15 ( 0.1 • Convolution Example |W(f)| 0.05 • -5 0 5 Convolution Properties 0.5 0.5 Frequency (Hz) • Parseval’s Theorem W (f) = a+i2π(f+F ) + a+i2π(f−F ) 0.5 • Energy Conservation 0 • Energy Spectrum a+i2πf -0.5 2 2 2 2 (rad/pi) Energy Spectrum • The units of |W (f)|2 are “energy per Hz” so that its integral, ∞ 2 , has units of energy. Ew = −∞ |W (f)| df 2 • The quantityR |W (f)| is called the energy spectral density of w(t) at frequency f and its graph is the energy spectrum of w(t). It shows how the energy of w(t) is distributed over frequencies. 2 • If you divide |W (f)| by the total energy, Ew, the result is non-negative and integrates to unity like a probability distribution. E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 9 / 10 Summary 7: Fourier Transforms: • Convolution: Convolution and Parseval’s ∞ Theorem ◦ u(t) ∗ v(t) , u(τ)v(t − τ)dτ • Multiplication of Signals −∞ • Multiplication Example • Convolution Theorem R • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 10 / 10 Summary 7: Fourier Transforms: • Convolution: Convolution and Parseval’s ∞ Theorem ◦ u(t) ∗ v(t) , u(τ)v(t − τ)dτ • Multiplication of Signals −∞ • Multiplication Example ⊲ Arguments of u(··· ) and v(··· ) sum to t • Convolution Theorem R • Convolution Example • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 10 / 10 Summary 7: Fourier Transforms: • Convolution: Convolution and Parseval’s ∞ Theorem ◦ u(t) ∗ v(t) , u(τ)v(t − τ)dτ • Multiplication of Signals −∞ • Multiplication Example ⊲ Arguments of u(··· ) and v(··· ) sum to t • Convolution Theorem R • Convolution Example ◦ Acts like multiplication + time scaling/shifting formulae • Convolution Properties • Parseval’s Theorem • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 10 / 10 Summary 7: Fourier Transforms: • Convolution: Convolution and Parseval’s ∞ Theorem ◦ u(t) ∗ v(t) , u(τ)v(t − τ)dτ • Multiplication of Signals −∞ • Multiplication Example ⊲ Arguments of u(··· ) and v(··· ) sum to t • Convolution Theorem R • Convolution Example ◦ Acts like multiplication + time scaling/shifting formulae • Convolution Properties • Parseval’s Theorem • Convolution Theorem: multiplication ↔ convolution • Energy Conservation • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 10 / 10 Summary 7: Fourier Transforms: • Convolution: Convolution and Parseval’s ∞ Theorem ◦ u(t) ∗ v(t) , u(τ)v(t − τ)dτ • Multiplication of Signals −∞ • Multiplication Example ⊲ Arguments of u(··· ) and v(··· ) sum to t • Convolution Theorem R • Convolution Example ◦ Acts like multiplication + time scaling/shifting formulae • Convolution Properties • Parseval’s Theorem • Convolution Theorem: multiplication ↔ convolution • Energy Conservation ◦ w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Energy Spectrum • Summary E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 10 / 10 Summary 7: Fourier Transforms: • Convolution: Convolution and Parseval’s ∞ Theorem ◦ u(t) ∗ v(t) , u(τ)v(t − τ)dτ • Multiplication of Signals −∞ • Multiplication Example ⊲ Arguments of u(··· ) and v(··· ) sum to t • Convolution Theorem R • Convolution Example ◦ Acts like multiplication + time scaling/shifting formulae • Convolution Properties • Parseval’s Theorem • Convolution Theorem: multiplication ↔ convolution • Energy Conservation ◦ w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Energy Spectrum • Summary ◦ w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 10 / 10 Summary 7: Fourier Transforms: • Convolution: Convolution and Parseval’s ∞ Theorem ◦ u(t) ∗ v(t) , u(τ)v(t − τ)dτ • Multiplication of Signals −∞ • Multiplication Example ⊲ Arguments of u(··· ) and v(··· ) sum to t • Convolution Theorem R • Convolution Example ◦ Acts like multiplication + time scaling/shifting formulae • Convolution Properties • Parseval’s Theorem • Convolution Theorem: multiplication ↔ convolution • Energy Conservation ◦ w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Energy Spectrum • Summary ◦ w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) ∞ +∞ Parseval’s Theorem: ∗ ∗ • t=−∞ u (t)v(t)dt = f=−∞ U (f)V (f)df R R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 10 / 10 Summary 7: Fourier Transforms: • Convolution: Convolution and Parseval’s ∞ Theorem ◦ u(t) ∗ v(t) , u(τ)v(t − τ)dτ • Multiplication of Signals −∞ • Multiplication Example ⊲ Arguments of u(··· ) and v(··· ) sum to t • Convolution Theorem R • Convolution Example ◦ Acts like multiplication + time scaling/shifting formulae • Convolution Properties • Parseval’s Theorem • Convolution Theorem: multiplication ↔ convolution • Energy Conservation ◦ w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Energy Spectrum • Summary ◦ w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) ∞ +∞ Parseval’s Theorem: ∗ ∗ • t=−∞ u (t)v(t)dt = f=−∞ U (f)V (f)df • Energy Spectrum: R R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 10 / 10 Summary 7: Fourier Transforms: • Convolution: Convolution and Parseval’s ∞ Theorem ◦ u(t) ∗ v(t) , u(τ)v(t − τ)dτ • Multiplication of Signals −∞ • Multiplication Example ⊲ Arguments of u(··· ) and v(··· ) sum to t • Convolution Theorem R • Convolution Example ◦ Acts like multiplication + time scaling/shifting formulae • Convolution Properties • Parseval’s Theorem • Convolution Theorem: multiplication ↔ convolution • Energy Conservation ◦ w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Energy Spectrum • Summary ◦ w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) ∞ +∞ Parseval’s Theorem: ∗ ∗ • t=−∞ u (t)v(t)dt = f=−∞ U (f)V (f)df • Energy Spectrum: R R ◦ Energy spectral density: |U(f)|2 (energy/Hz) E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 10 / 10 Summary 7: Fourier Transforms: • Convolution: Convolution and Parseval’s ∞ Theorem ◦ u(t) ∗ v(t) , u(τ)v(t − τ)dτ • Multiplication of Signals −∞ • Multiplication Example ⊲ Arguments of u(··· ) and v(··· ) sum to t • Convolution Theorem R • Convolution Example ◦ Acts like multiplication + time scaling/shifting formulae • Convolution Properties • Parseval’s Theorem • Convolution Theorem: multiplication ↔ convolution • Energy Conservation ◦ w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Energy Spectrum • Summary ◦ w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) ∞ +∞ Parseval’s Theorem: ∗ ∗ • t=−∞ u (t)v(t)dt = f=−∞ U (f)V (f)df • Energy Spectrum: R R ◦ Energy spectral density: |U(f)|2 (energy/Hz) 2 2 ◦ Parseval: Eu = |u(t)| dt = |U(f)| df R R E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 10 / 10 Summary 7: Fourier Transforms: • Convolution: Convolution and Parseval’s ∞ Theorem ◦ u(t) ∗ v(t) , u(τ)v(t − τ)dτ • Multiplication of Signals −∞ • Multiplication Example ⊲ Arguments of u(··· ) and v(··· ) sum to t • Convolution Theorem R • Convolution Example ◦ Acts like multiplication + time scaling/shifting formulae • Convolution Properties • Parseval’s Theorem • Convolution Theorem: multiplication ↔ convolution • Energy Conservation ◦ w(t) = u(t)v(t) ⇔ W (f) = U(f) ∗ V (f) • Energy Spectrum • Summary ◦ w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) ∞ +∞ Parseval’s Theorem: ∗ ∗ • t=−∞ u (t)v(t)dt = f=−∞ U (f)V (f)df • Energy Spectrum: R R ◦ Energy spectral density: |U(f)|2 (energy/Hz) 2 2 ◦ Parseval: Eu = |u(t)| dt = |U(f)| df R For furtherR details see RHB Chapter 13.1 E1.10FourierSeriesandTransforms(2014-5559) FourierTransform - Parseval and Convolution: 7 – 10 / 10