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: