Fourier Transforms of Discrete Signals

7/29/2019 Fourier Transforms of Discrete Signals

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Chapter 12

Fourier Transforms of Discrete Signals


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Sampling

Continuous signals are digitized using digital computers

When we sample, we calculate the value of the continuoussignal 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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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 CDshould sample at the rate of 44.1 KHz


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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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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


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Example D

See map!


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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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Finding DTFT For periodic signals

Starting with xo[n]

DTFT of xo[n]

Example


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Example A & B

notes

notes

X[n]=a|n|, 0 < a < 1.


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DT Fourier Transforms

1. W is in radian and it is between

0 and 2p in each discrete time

interval

2. This is different from w where it

was between INF and + INF

3. Note that X(W) is periodic


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Properties of DTFT

Remember:

For time scaling note that

m>1 Signal spreading


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Fourier Transform of Periodic Sequences

Check the map~~~~~See map!


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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 (Npoints)


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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 is2pi/N.

We can think of DFT as one period of discrete Fourier series


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A short hand notation

remember:


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Inverse of DFT

We can obtain the inverse of DFT

Note that


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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


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Example of DFT

Find X[k]

We know k=1,.., 7; N=8


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Example of DFT


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Example of DFT

Time shift Property of DFT

Other DFT properties: http://cnx.org/content/m12019/latest/

Polar plot for


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Example of DFT

E l f DFT


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Example of DFT

Summation for X[k]

Using the shift property!

E l f IDFT


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Example of IDFT

Remember:

E l f IDFT


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Example of IDFT

Remember:

F t F i T f Al ith


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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

R di 2 FFT Al ith T i t FFT


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Radix-2 FFT Algorithms - Two point FFT

We assume N=2^m

This is called Radix-2 FFT Algorithms

Lets 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)

G l FFT Al ith


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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 letting2mm

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

jNN

m

N

N

N

m

N

Nm

N

mk

N

mk

N

N

m

mkN

kN

N

m

mkN

W

jeW

WWWW

WW

mxWWmxW

ppp

E l


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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

mkN

kN

N

m

mkN

W

jeW

WWWW

WW

mxWWmxWkX

ppp

Lets 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

FFT Al ith F i t FFT


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FFT Algorithms - Four point FFT

First find even and odd parts and then combine them:

The general form:

FFT Algorithms 8 point FFT


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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


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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


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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


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References

Read Schaums Outlines: Chapter 6

Do Chapter 12 problems: 19, 20, 26, 5, 7

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Fourier Transforms of Discrete Signals