1.
Fourier Transforms of Discrete
Signals
Microlink IT College
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2.
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
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3.
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
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4.
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):
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5.
Discrete Time Fourier Transform
• Discrete-Time Fourier Transform (DTFT):
• A few points
– DTFT is periodic in frequency with period of 2π
– X[n] is a discrete signal
– DTFT allows us to find the spectrum of the discrete signal as viewed
from a window
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7.
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!
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8.
Finding DTFT For periodic signals
• Starting with xo[n]
• DTFT of xo[n]
Example
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9.
Example A & B
notes
notes
X[n]=a|n|
, 0 < a < 1.
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10.
DT Fourier Transforms
1. Ω is in radian and it is between
0 and 2π in each discrete time
interval
2. This is different from ω where it
was between – INF and + INF
3. Note that X(Ω) is periodic
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11.
Properties of DTFT
• Remember:
• For time scaling note that
m>1 Signal spreading
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12.
Fourier Transform of Periodic Sequences
• Check the map~~~~~
See map!
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13.
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)
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14.
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
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25.
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
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26.
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)
Advantage: Less
computationally
intensive: N/2.log(N)
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27.
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/
0
2/
12/
0
2/
−=
=−−−==
==
=
++
−
+
−
=
−
=
∑∑
N
N
jN
N
m
N
N
N
m
N
Nm
N
mk
N
mk
N
N
m
mk
N
k
N
N
m
mk
N
W
jeW
WWWW
WW
mxWWmxW
πππ
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28.
Example
1
1)2sin()2cos(
)]12[(]2[][
2/
2
2/
2/
2/2/
2/
2/
2/
2
12/
0
2/
12/
0
2/
−=
=−−−==
==
=
++=
−
+
−
=
−
=
∑∑
N
N
jN
N
m
N
N
N
m
N
Nm
N
mk
N
mk
N
N
m
mk
N
k
N
N
m
mk
N
W
jeW
WWWW
WW
mxWWmxWkX
πππ
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[
1
1
0
0
1.0
1
1
1
0
0
1.0
1
0
0
0.0
1
0
1
0
0
0.0
1
xxxWxxWWxWkX
xxxWWxWkX
mm
mm
−=+=+==
+=+==
∑∑
∑∑
==
==
Same result
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29.
FFT Algorithms - Four point FFT
First find even and odd parts and then combine them:
The general form:
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31.
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!
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32.
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]}.
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