Chapter 12 Fourier Transforms of Discrete Signals

Sampling Continuous signals are digitized using digital computers When we sample, we calculate the value of the continuous signal at discrete points –How fast do we sample –What is the value of each point Quantization determines the value of each samples value

Sampling Periodic Functions - Note that wb = Bandwidth, thus if then aliasing occurs (signal overlaps) -To avoid aliasing -According sampling theory: To hear music up to 20KHz a CD should sample at the rate of 44.1 KHz

Discrete Time Fourier Transform In likely we only have access to finite amount of data sequences (after sampling) Recall for continuous time Fourier transform, when the signal is sampled: Assuming Discrete-Time Fourier Transform (DTFT):

Discrete Time Fourier Transform Discrete-Time Fourier Transform (DTFT): A few points –DTFT is periodic in frequency with period of 2  –X[n] is a discrete signal –DTFT allows us to find the spectrum of the discrete signal as viewed from a window

Example D See map!

Example of Convolution Convolution –We can write x[n] (a periodic function) as an infinite sum of the function x o [n] (a non-periodic function) shifted N units at a time –This will result –Thus See map!

Finding DTFT For periodic signals Starting with xo[n] DTFT of xo[n] Example

Example A & B notes X[n]=a |n|, 0 < a < 1.

DT Fourier Transforms  is in radian and it is between 0 and 2  in each discrete time interval 2.This is different from  where it was between – INF and + INF 3.Note that X(  ) is periodic

Properties of DTFT Remember: For time scaling note that m>1  Signal spreading

Fourier Transform of Periodic Sequences Check the map~~~~~ See map!

Discrete Fourier Transform We often do not have an infinite amount of data which is required by DTFT –For example in a computer we cannot calculate uncountable infinite (continuum) of frequencies as required by DTFT Thus, we use DTF to look at finite segment of data –We only observe the data through a window –In this case the xo[n] is just a sampled data between n=0, n=N-1 (N points)

Discrete Fourier Transform It turns out that DFT can be defined as Note that in this case the points are spaced 2pi/N; thus the resolution of the samples of the frequency spectrum is 2pi/N. We can think of DFT as one period of discrete Fourier series

A short hand notation remember:

Inverse of DFT We can obtain the inverse of DFT Note that

Using MATLAB to Calculate DFT Example: –Assume N=4 –x[n]=[1,2,3,4] –n=0,…,3 –Find X[k]; k=0,…,3 or

Example of DFT Find X[k] –We know k=1,.., 7; N=8

Example of DFT

Time shift Property of DFT Other DFT properties: Polar plot for

Example of DFT

Summation for X[k] Using the shift property!

Example of IDFT Remember:

Example of IDFT Remember:

Fast Fourier Transform Algorithms Consider DTFT Basic idea is to split the sum into 2 subsequences of length N/2 and continue all the way down until you have N/2 subsequences of length 2 Log2(8) N

Radix-2 FFT Algorithms - Two point FFT We assume N=2^m –This is called Radix-2 FFT Algorithms Let’s take a simple example where only two points are given n=0, n=1; N=2 y0 y1 y0 Butterfly FFT Advantage: Less computationally intensive: N/2.log(N)

General FFT Algorithm First break x[n] into even and odd Let n=2m for even and n=2m+1 for odd Even and odd parts are both DFT of a N/2 point sequence Break up the size N/2 subsequent in half by letting 2m  m The first subsequence here is the term x[0], x[4], … The second subsequent is x[2], x[6], …

Example Let’s take a simple example where only two points are given n=0, n=1; N=2 Same result

FFT Algorithms - Four point FFT First find even and odd parts and then combine them: The general form:

FFT Algorithms - 8 point FFT Applet:

A Simple Application for FFT t = 0:0.001:0.6; % 600 points x = sin(2*pi*50*t)+sin(2*pi*120*t); y = x + 2*randn(size(t)); plot(1000*t(1:50),y(1:50)) title('Signal Corrupted with Zero-Mean Random Noise') xlabel('time (milliseconds)') Taking the 512-point fast Fourier transform (FFT): Y = fft(y,512) The power spectrum, a measurement of the power at various frequencies, is Pyy = Y.* conj(Y) / 512; Graph the first 257 points (the other 255 points are redundant) on a meaningful frequency axis: f = 1000*(0:256)/512; plot(f,Pyy(1:257)) title('Frequency content of y') xlabel('frequency (Hz)') This helps us to design an effective filter! ML Help!

Example Express the DFT of the 9-point {x[0], …,x[9]} in terms of the DFTs of 3-point sequences {x[0],x[3],x[6]}, {x[1],x[4],x[7]}, and {x[2],x[5],x[8]}. Later

References Read Schaum’s Outlines: Chapter 6 Do Chapter 12 problems: 19, 20, 26, 5, 7