Fourier Transforms of Discrete Signals By Dr. Varsha Shah RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP 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 RCOE Dr. Varsha Shah DSPP
Sampling Periodic Functions To hear music up to 20KHz a CD should sample at the rate of 44.1 KHz RCOE Dr. Varsha Shah DSPP
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): RCOE Dr. Varsha Shah DSPP
Discrete Time Fourier Transform Discrete-Time Fourier Transform (DTFT): A few points DTFT is periodic in frequency with period of 2p X[n] is a discrete signal DTFT allows us to find the spectrum of the discrete signal as viewed from a window RCOE Dr. Varsha Shah DSPP
Example of Convolution We can write x[n] (a periodic function) as an infinite sum of the function xo[n] (a non-periodic function) shifted N units at a time This will result Thus RCOE Dr. Varsha Shah DSPP
Finding DTFT For periodic signals Starting with Xo [n] DTFT of Xo [n] RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP DT Fourier Transforms W is in radian and it is between 0 and 2p in each discrete time interval This is different from w where it was between – INF and + INF Note that X(W) is periodic RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP Properties of DTFT Remember: For time scaling note that m>1 Signal spreading RCOE Dr. Varsha Shah DSPP
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) RCOE Dr. Varsha Shah DSPP
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 RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP A short hand notation remember: RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP Inverse of DFT We can obtain the inverse of DFT Note that RCOE Dr. Varsha Shah DSPP
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 RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP Example of DFT Find X[k] We know k=1,.., 7; N=8 RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP Example of DFT RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP Example of DFT Polar plot for Time shift Property of DFT RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP Example of DFT RCOE Dr. Varsha Shah DSPP
Example of DFT Summation for X[k] Using the shift property! RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP Example of IDFT Remember: RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP Example of IDFT Remember: RCOE Dr. Varsha Shah DSPP
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 RCOE Dr. Varsha Shah DSPP
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 Butterfly FFT y0 y0 y1 Advantage: Less computationally intensive: N/2.log(N) RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP 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 2mm The first subsequence here is the term x[0], x[4], … The second subsequent is x[2], x[6], … RCOE Dr. Varsha Shah DSPP
RCOE Dr. Varsha Shah DSPP Example Let’s take a simple example where only two points are given n=0, n=1; N=2 Same result RCOE Dr. Varsha Shah DSPP
FFT Algorithms - Four point FFT First find even and odd parts and then combine them: The general form: RCOE Dr. Varsha Shah DSPP
FFT Algorithms - 8 point FFT RCOE Dr. Varsha Shah DSPP
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! RCOE Dr. Varsha Shah DSPP