Lecture 7

Lecture 7: Continuous and Discrete Fourier Transforms and Convolution Learning Objectives: • Review of continuous and discrete Fourier transform (DFT) properties • Review examples of leakage and symmetry properties of the DFT • Learn built-in Fourier transform functions in Matlab • Extend these concepts to 2 dimensions (2D)

Assignment: 1. Read Chapter 2 of Bracewell, “The Fourier Transform and its Applications,” titled “Groundwork.” 2. Read Chapter 11 of Bracewell, “The Fourier Transform and its Applications,” titled “The Discrete Fourier Transform and the FFT.”

I. 1D continuous Fourier transform

F (ξ ) =

∞

∫ f ( x) e

−i 2πξx

dx

−∞ ∞

f ( x) =

∫ F (ξ ) e

i 2πξx

dx

−∞

Key FT pairs: cos(2πxξ ' ) sin(2πxξ ' ) rect (x)

δ (x)

⇔ ⇔ ⇔

f (ax)

⇔ ⇔ ⇔

comb( ∆xx )

⇔

δ ( x − x ′)

Symmetry Properties: Real and even Real and Assymetric Even Odd ∞

F (0) =

∫ f ( x) dx

−∞

⇔ ⇔ ⇔ ⇔ ⇔

1 2

i 2

{δ (ξ − ξ ) + δ (ξ − ξ )} {δ (ξ + ξ ) − δ (ξ − ξ )} '

'

'

'

sinc (ξ ) =

sin(πξ )

πξ

1 −2πx ′ξ

⋅1 1 a F(a) ∆x comb(∆xξ ) e

ξ

Real and even Complex and Hermetian F (ξ ) = F * (−ξ ) Even Odd ∞

f (0) =

∫ F (ξ ) dξ

−∞

Lecture 7: Continuous and Discrete Fourier Transforms and Convolution B. Discrete Fourier Transform Figure 1:

(After Bracewell) N −1

F (k ) = ∆x ∑ f (n)e

− i 2π

nk N

n=0

N −1

f (n) = ∆ξ ∑ F (k )e

i 2π

nk N

k =0

1. Now talking about a periodic function of discrete points a. In general symmetry properties all apply to DFT as well b. Periodic nature of the DFT has consequences not encountered in the continuous FT f ( x) ⋅ comb( ∆xx )

⇔

F (ξ ) ∗ ∆x comb(∆xξ )

Periodicity

f (n∆x) ⋅ rect ( Lx )

⇔

F (k∆ξ ) ∗ L sinc( Lξ )

Compact Support

Sampling Demo (http://users.ece.gatech.edu/mcclella/matlabGUIs/index.html):

Lecture 7: Continuous and Discrete Fourier Transforms and Convolution 1. Interval (or window) of evaluation (N = 1000 samples; fs = 1000 Hz; signals: 60, 65 and 70 Hz): Causal window: t = 0:dt:(n-1)*dt;

Shifted window: t = -(n-1)/2*dt:dt:(n-1)/2*dt;

Lecture 7: Continuous and Discrete Fourier Transforms and Convolution Causal window: Non-integer number of wavelength width (n = 350) - leakage is observed.

f ( x) ⋅ comb( ∆xx )

⇔

F (ξ ) ∗ ∆x comb(∆xξ )

Periodicity

f (n∆x) ⋅ rect ( Lx )

⇔

F (k∆ξ ) ∗ L sinc( Lξ )

Compact Support

(Ludeman, “Digital Signal Processing”)

Lecture 7: Continuous and Discrete Fourier Transforms and Convolution 2. DC term and computer representation: a. Shift theorem b. Convolution theorem 1. Continuous vs. circular convolution a. length N1+N2-1 vs. length N:

(from Bracewell) C = conv(A, B); % zeropads so that the convolution is length(A)+length(B)-1

1D FT - fftshift

1D FT - 2xfftshift

Use fftshift

12

abs (L2)

12 10 8 6 4

1

2

0.9

0

0.8

10

20

30

40

50

60

Ê

0.7

4

l

phase (L1)

0.6 0.5 0.4 0.3 0.2 0.1 0

1 0

2 0 -2 -4

10

20

30

40

33 sample # x [Matlab]

50

60

10

20

30

40

50

60

64

1

33 sample #

64

xmax

0

−Ν/2 ∆k

− ∆k

k

>>L2 = fftshift(fft(fftshift(l))); fftshift(fft(fftshift(l)));

10 8 6 4 2

Center Fourier Domain

0

10

20

30

40

50

60

10

20

30

40

50

60

4

phase (L2)

>>L1 = fft(fftshift(l)); fft(fftshift(l));

abs (L1)

Ê

2 0 -2 -4

1 - N/2 ∆k

33 sample # 0

64 (N-1)/2 ∆k

Lecture 7: Continuous and Discrete Fourier Transforms and Convolution 3. Scaling >> x = ones(1,64) >> X = fftshift(fft(fftshift(x))); >> figure;stem(abs(X)) %Figure A

>> x_back = fftshift(ifft(fftshift(X))); >> figure;plot(abs(x_back)) %Figure B

>> x_back = fftshift(fft(fftshift(X))); >> figure;plot(abs(x_back)) %Figure C