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The Transform: Examples, Properties, Common Pairs The : Examples, Properties, Common Pairs

Magnitude and

Remember: complex numbers can be thought of as (real,imaginary) The Fourier Transform: or (magnitude,phase). Examples, Properties, Common Pairs 2 2 1/2 CS 450: Introduction to Digital Signal and Image Processing Magnitude: F = (F) + (F) | | < 1 (F)= = Phase: φ (F) = tan− (F)  < 

Bryan Morse Real part How much of a cosine of that you need BYU Computer Science Imaginary part How much of a of that frequency you need Magnitude of combined cosine and sine Phase Relative proportions of sine and cosine

The Fourier Transform: Examples, Properties, Common Pairs The Fourier Transform: Examples, Properties, Common Pairs

Example: Fourier Transform of a Cosine Example: Fourier Transform of a Cosine

( ) = ( π ) f t cos 2 st Spatial Domain ∞ i2πut F(u) = f (t) e− dt 1 1 Z−∞ cos(2πst) 2 δ(u s) + 2 δ(u + s) ∞ i2πut − = cos(2πst) e− dt 1 Z−∞ 1 ∞ = cos(2πst)[cos( 2πut) + i sin( 2πut)] dt 0.8 − − 0.5 Z−∞ 0.6 ∞ ∞ = cos(2πst) cos( 2πut) dt + i cos(2πst) sin( 2πut) dt 0.2 0.4 0.6 0.8 1 − − 0.4 Z−∞ Z−∞ -0.5 ∞ ∞ 0.2 = cos(2πst) cos(2πut) dt i cos(2πst) sin(2πut) dt − Z−∞ Z−∞ -1 -10 -5 5 10 0 except when u = s 0 for all u ± 1 1 = δ(u s) + δ(u + s) 2 − 2

The Fourier Transform: Examples, Properties, Common Pairs The Fourier Transform: Examples, Properties, Common Pairs

Odd and Even Functions Sinusoids

Even Odd Spatial Domain Frequency Domain f ( t) = f (t) f ( t) = f (t) − − − f (t) F(u) Symmetric Anti-symmetric Cosines 1 cos(2πst) 2 [δ(u + s) + δ(u s)] Transform is real∗ Transform is imaginary∗ − sin(2πst) 1 i [δ(u + s) δ(u s)] 2 − −

∗ for real-valued signals The Fourier Transform: Examples, Properties, Common Pairs The Fourier Transform: Examples, Properties, Common Pairs

Constant Functions Delta Functions

Spatial Domain Frequency Domain Spatial Domain Frequency Domain f (t) F(u) f (t) F(u) 1 δ(u) δ(t) 1 a a δ(u)

The Fourier Transform: Examples, Properties, Common Pairs The Fourier Transform: Examples, Properties, Common Pairs

Square Pulse Square Pulse

Spatial Domain Frequency Domain f (t) F(u)

1 if a/2 t a/2 ( π ) − ≤ ≤ sinc(aπu) = sin a u 0 otherwise aπu 

The Fourier Transform: Examples, Properties, Common Pairs The Fourier Transform: Examples, Properties, Common Pairs

Triangle Comb

Spatial Domain Frequency Domain Spatial Domain Frequency Domain f (t) F(u) f (t) F(u)

1 t if a t a − | | − ≤ ≤ sinc2(aπu) δ(t mod k) δ(u mod 1/k) 0 otherwise  The Fourier Transform: Examples, Properties, Common Pairs The Fourier Transform: Examples, Properties, Common Pairs

Gaussian Differentiation

Spatial Domain Frequency Domain Spatial Domain Frequency Domain f (t) F(u) f (t) F(u)

πt2 πu2 d e− e− dt 2πiu

The Fourier Transform: Examples, Properties, Common Pairs The Fourier Transform: Examples, Properties, Common Pairs

Some Common Fourier Transform Pairs More Common Fourier Transform Pairs

Spatial Domain Frequency Domain Spatial Domain Frequency Domain f (t) F(u) f (t) F(u) Cosine cos(2πst) Deltas 1 [δ(u + s) + δ(u s)] 1 if a/2 t a/2 2 − Square − ≤ ≤ Sinc sinc(aπu) Sine sin(2πst) Deltas 1 i [δ(u + s) δ(u s)] 0 otherwise 2 − − Unit 1 Delta δ(u) 1 t if a t a 2 Triangle − | | − ≤ ≤ Sinc2 sinc (aπu) a Delta aδ(u) 0 otherwise πt2 πu2 Delta δ(t) Unit 1 Gaussian e− Gaussian e− d Comb δ(t mod k) Comb δ(u mod 1/k) Differentiation dt Ramp 2πiu

The Fourier Transform: Examples, Properties, Common Pairs The Fourier Transform: Examples, Properties, Common Pairs

Properties: Notation Properties: Linearity

Let denote the Fourier Transform: Adding two functions together adds their Fourier Transforms together: F F = (f ) (f + g) = (f ) + (g) F F F F 1 Multiplying a by a scalar constant multiplies its Fourier Let − denote the Inverse Fourier Transform: F Transform by the same constant: 1 f = − (F) F (af ) = a (f ) F F The Fourier Transform: Examples, Properties, Common Pairs The Fourier Transform: Examples, Properties, Common Pairs

Properties: Translation Change of Scale: Square Pulse Revisited

Translating a function leaves the magnitude unchanged and adds a constant to the phase. If f = f (t a) 2 1 − F = (f ) 1 F 1 F = (f ) 2 F 2 then F = F | 2| | 1| φ (F ) = φ (F ) 2πua 2 1 − Intuition: magnitude tells you “how much”, phase tells you “where”.

The Fourier Transform: Examples, Properties, Common Pairs

Rayleigh’s Theorem

Total “” (sum of squares) is the same in either domain:

∞ ∞ f (t) 2 dt = F(u) 2 du | | | | Z−∞ Z−∞