Lesson 4

Lesson 4Discrete Fourier Transform

How to sample the frequency axis?

Sampling of Analog Signals

Aliasing Formula

How to sample the frequency axis?

Fourier Series of a Periodic Sequence

Fourier Series of a Periodic Sequence

Fourier Series of a Periodic Sequence

Matlab Implementation

Analysis or DFS equation

Synthesis or inverse DFS equation

Matlab Implementation

Relation to DTFT

sinc function

Periodic Signals Are Completely Discrete:

Discrete rather than continuous frequencies. Discrete rather than continuous times. Summations instead of integrals.

Aperiodic Discrete-Time Signals

Aperiodic Discrete-Time Signals

Sampling in frequency generates a periodic signal in time

DFT of an Aperiodic Discrete-Time Signal of length N

Choose an integer L larger than or equal to N to be the period of a periodic extension of the aperiodic signal x(n). Pad zeros to x(n) if necessary.

Find the normalized Fourier Series representation of the periodic extension through DFS.

Then the DFT of x(n) is given by the DFS of the periodic extension for k on [0 L-1] and the IDFT is given by the IDFS with n on [0 L-1].

Computation of DFT via FFT

FFT (Fast Fourier Transform) is not another transformation but an algorithm to efficiently compute DFT.

Causal aperiodic signals: {x(n), n = 0, 1, … N-1}: proceed using FFT to obtain {X(k), k=0,1,…, N-1}. To compute for L>N, we simply attach L-N zeros at the end of the x(n) sequence and then FFT to obtain L values.

Computation of DFT via FFT

Non-Causal aperiodic signals: {x(n), n=-n0, …, 0, 1, …, N-n0-1}: Move the non-causal samples to the causal side: {x(0),

x(1), …, x(N-n0-1), x(-n0), x(-n0+1),…,x(-1)} To improve frequency resolution, attach zeros between

the causal and non-causal samples: {x(0), x(1), …, x(N-n0-1), 0, 0, 0, …, 0, 0, 0, x(-n0), x(-n0+1),…,x(-1)}

Example

Consider the DFT computation via FFT of a causal signalx(n) = (sin(πn/32))(u(n)-u(n-34)) and its shifted version x(n+16). To improve its frequency resolution, compute FFTs of length N = 512.

Convolution

Convolution

Convolution

DTFT of y

Convolution

We can obtain y(n) through the inverse Fourier transform

The L-length DFT of x(n) and h(n) are obtained by padding zeros. Pad x(n) with L-M zeros Pad h(n) with L-K zeros

Convolution

Given x(n) and h(n) of lengths M and K, the convolution y(n) of length N=M+K-1 can be found by the following 3 steps: Compute DFTs X(k) and H(k) of length L>=N for x(n) and

h(n). Multiply them to get Y(k)=X(k)H(k) Find the inverse DFT of Y(k) of length L to obtain y(n)

Example

x(n) = u(n) – u(n-21) of length 20, convolve with itself for different values of its length.