chapter 12 fourier transforms of discrete signals
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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 xo[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
notes
X[n]=a|n|, 0 < a < 1.
DT Fourier Transforms
1. 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
Example of DFT
Time shift Property of DFT
Other DFT properties: http://cnx.org/content/m12019/latest/
Polar plot for
Example of DFT
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
http://www.cmlab.csie.ntu.edu.tw/cml/dsp/training/coding/transform/fft.html
y0
y1
y0
Butterfly FFT
Advantage: Less computationally intensive: N/2.log(N)
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 2mm
• The first subsequence here is the term x[0], x[4], …• The second subsequent is x[2], x[6], …
1
1)2sin()2cos(
)]12[(]2[
2/
2
2/2/
2/2/2/
2/
2/2
12/
02/
12/
02/
NN
jNN
mN
NN
mN
NmN
mkN
mkN
N
m
mkN
kN
N
m
mkN
W
jeW
WWWW
WW
mxWWmxW
Example
1
1)2sin()2cos(
)]12[(]2[][
2/
2
2/2/
2/2/2/
2/
2/2
12/
02/
12/
02/
NN
jNN
mN
NN
mN
NmN
mkN
mkN
N
m
mkN
kN
N
m
mkN
W
jeW
WWWW
WW
mxWWmxWkX
Let’s take a simple example where only two points are given n=0, n=1; N=2
]1[]0[]1[]0[)]1[(]0[]1[
]1[]0[)]1[(]0[]0[
11
0
0
1.01
11
0
0
1.01
0
0
0.01
01
0
0
0.01
xxxWxxWWxWkX
xxxWWxWkX
mm
mm
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
http://www.engineeringproductivitytools.com/stuff/T0001/PT07.HTM
Applet: http://www.falstad.com/fourier/directions.html
A Simple Application for FFT
t = 0:0.001:0.6; % 600 pointsx = 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
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