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