# Chapter 1-10 Notes on DSP, Study notes for Digital Signal Processing

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EE-2200 Fall-98

1

ECE-2026 Spring-2018

Lecture 9

Sampling & Aliasing

12-Feb-2018

2/4/2018 ECE2026, Spring 2018 2

General Info

 ITS Questions: ITS #2 due today  Full credit with 2400 points per ITS session

 Lab #04: start this Week: Spectrogram, AM, FM  Lab #05 next week: Fourier Series

 HW #04 due this Week

 Quiz #1 score adjustment: before 2/23 (Friday)

 #1: Dr. Fekri (faramarz.fekri@ece.gatech. edu)

 #2: Dr. Yang (Benjamin.Yang@gtri.gatech.edu)

 #3: Dr. Marenco (Alvaro.Marenco@gtri.gatech.edu)

2

2/4/2018 3

Preliminary Quiz #1 Summary

Mean: 57.9

Median: 56

StdDev: 19.5

 An example summary from spring 2017

ECE2026, Spring 2018

Overlapping Sections in

Spectrograms (Labs 4 & 5)

 50% overlap is common

 Consider edge effects when analyzing a short sinusoid

2/4/2018 ECE2026, Spring 2018 4

SECTION

LOCATIONS

MIDDLE of SECTION

is REFERENCE TIME

3

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 This Lecture:

 Chap 4, Sections 4-1 and 4-2

 Replaces Ch 4 in DSP First, pp. 83-94

 Recitation: Strobe Demo (Sect 4-3)

 Next Lecture: Chap. 4 Sects. 4-4 and 4-5

2/4/2018 ECE2026, Spring 2018 6

LECTURE OBJECTIVES

 SAMPLING can cause ALIASING

Sampling Theorem

 Sampling Rate > 2*(highest frequency term)

 Spectrum for digital signals, x[n]

 Normalized Frequency

 

 2 2

ˆ  s

s f

f T

ALIASING

4

2/4/2018 ECE2026, Spring 2018 7

SYSTEMS Process Signals

 PROCESSING GOALS:

 Change x(t) into y(t)

 For example, more BASS, pitch shifting

 Improve x(t), e.g., image deblurring

 Extract Information from x(t)

SYSTEM x(t) y(t)

2/4/2018 ECE2026, Spring 2018 8

System IMPLEMENTATION

 DIGITAL/MICROPROCESSOR  Convert x(t) to numbers stored in memory

ELECTRONICS x(t) y(t)

COMPUTER D-to-AA-to-D x(t) y(t)y[n]x[n]

 ANALOG/ELECTRONIC:  Circuits: resistors, capacitors, op-amps

5

2/4/2018 ECE2026, Spring 2018 9

SAMPLING x(t)

 SAMPLING PROCESS  Convert x(t) to numbers x[n]

 “n” is an integer; x[n] is a sequence of values

 Think of “n” as the storage address in memory

 UNIFORM SAMPLING at t = nTs  IDEAL: x[n] = x(nTs)

C-to-D x(t) x[n]

2/4/2018 ECE2026, Spring 2018 10

SAMPLING RATE, f s

 SAMPLING RATE (fs)

fs =1/Ts : in MATLAB - tt=0:1/8000:5.0  NUMBER of SAMPLES PER SECOND

Ts = 125 microsec  fs = 8000 samples/sec • UNITS ARE HERTZ: 8000 Hz

 UNIFORM SAMPLING at t = nTs = n/fs  IDEAL: x[n] = x(nTs)=x(n/fs)

C-to-D x(t) x[n]=x(nTs)

6

2/4/2018 ECE2026, Spring 2018 11

STORING DIGITAL SOUND

x[n] is a SAMPLED SINUSOID

 A list of numbers stored in memory

 EXAMPLE: audio CD

 CD rate is 44,100 samples per second

 16-bit samples

 Stereo uses 2 channels

 Number of bytes for 1 minute is

 2 X (16/8) X 60 X 44100 = 10.584 Mbytes

Be careful…

2/4/2018 ECE2026, Spring 2018 12

7

2/4/2018 ECE2026, Spring 2018 13

fs  2 kHz

fs  500Hz

Hz100f

Which one provides most accurate representation of x(t)?

2/4/2018 ECE2026, Spring 2018 14

SAMPLING THEOREM

 HOW OFTEN DO WE NEED TO SAMPLE?

 DEPENDS on FREQUENCY of SINUSOID

 ALSO DEPENDS on “RECONSTRUCTION

8

2/4/2018 ECE2026, Spring 2018 15

Reconstruction? Which One?

)4.0cos(][ nnx 



When n is an integer

cos(0.4n)  cos(2.4n)

Given the samples, draw a sinusoid through the values

Occam’s razor -> pick lowest frequency sinusoid

2/4/2018 ECE2026, Spring 2018 16

Spatial Aliasing (Lab6)

9

2/4/2018 ECE2026, Spring 2018 17

Choice for Lab6 Report)

2/4/2018 ECE2026, Spring 2018 18

sfs T

nAnx 







ˆ

)ˆcos(][

)cos()(][

)cos()(









ss nTAnTxnx

tAtx

DISCRETE-TIME SINUSOID

 Change x(t) into x[n] DERIVATION

))cos((][   nTAnx s

DEFINE DIGITAL FREQUENCY

10

2/4/2018 ECE2026, Spring 2018 19

DIGITAL FREQUENCY

 VARIES from 0 to 2, as f varies from

0 to the sampling frequency

 DIGITAL FREQUENCY is NORMALIZED



ˆ Ts  2f

fs



ˆ 

̂

2/4/2018 ECE2026, Spring 2018 20

SPECTRUM (DIGITAL)

sf

f  2ˆ 

kHz1sf ˆ 

1 2 X

1 2 X

*

2(0.1)–0.2

))1000/)(100(2cos(][   nAnx

11

2/4/2018 ECE2026, Spring 2018 21

SPECTRUM (DIGITAL) ???



ˆ 

1 2 X

1 2 X

*

2(1)–2

?

x[n] is zero frequency???

))100/)(100(2cos(][   nAnx

sf

f  2ˆ 



fs 100 Hz

2/4/2018 ECE2026, Spring 2018 22

The REST of the STORY

 Spectrum of x[n] has more than one line for

each complex exponential

 Called ALIASING

MANY SPECTRAL LINES

 SPECTRUM is PERIODIC with period = 2  Because

( ) ( )( )  nAnA 2ˆcosˆcos

12

2/4/2018 ECE2026, Spring 2018 23

ALIASING DERIVATION

 Other Frequencies give the same

s

s f

f T

 

2 ˆ  2



ˆ 

s

s

ss

s

f

f

f

f

f

ff   

22)(2 ˆ :then 

 



and we want : x[n] Acos( ˆ n)

))(2cos()( If   tffAtx s



tn

fs

2/4/2018 ECE2026, Spring 2018 24

ALIASING DERIVATION

 Other Frequencies give the same

Hz1000at sampled)400cos()(1  sfttx

)4.0cos()400cos(][ 10001

nnx n  

Hz1000at sampled)2400cos()(2  sfttx

)4.2cos()2400cos(][ 10002

nnx n  

)4.0cos()24.0cos()4.2cos(][2 nnnnnx  

][][ 12 nxnx  )1000(24002400  



ˆ 

13

2/4/2018 ECE2026, Spring 2018 25

ALIASING CONCLUSIONS

 Adding an INTEGER multiple of fs or –fs to

the frequency of a continuous sinusoid xc(t)

gives exactly the same values for the

sampled signal x[n] = xc(n/fs )

GIVEN x[n], we CAN’T KNOW whether it came

from a sinusoid at fo or (fo + fs ) or (fo + 2fs ) …

This is called ALIASING

2/4/2018 ECE2026, Spring 2018 26

SPECTRUM for x[n]

 PLOT versus NORMALIZED FREQUENCY

 INCLUDE ALL SPECTRUM LINES

 ALIASES

 SUBTRACT MULTIPLES of 2

 FOLDED ALIASES

 (to be discussed later)

 ALIASES of NEGATIVE FREQS

14

2/4/2018 ECE2026, Spring 2018 27

SPECTRUM (MORE LINES)

ˆ 

1 2 X

1 2 X

*

2(0.1)–0.2

1 2 X

*

1.8

1 2 X

–1.8

))1000/)(100(2cos(][   nAnx

kHz1sf

sf

f  2ˆ 

2/4/2018 ECE2026, Spring 2018 28

SPECTRUM (ALIASING CASE)

1 2 X

*

–0.5

1 2 X

–1.5

1 2 X

0.52.5–2.5 ˆ 

1 2 X

1 2 X

* 1 2 X

*

1.5

))80/)(100(2cos(][   nAnx

kHz80sf

sf

f  2ˆ 

15

2/4/2018 ECE2026, Spring 2018 29

SAMPLING GUI (con2dis, Lab6)

2/4/2018 ECE2026, Spring 2018 30

SPECTRUM (FOLDING CASE)

ˆ  2 f

fs

fs  125Hz

1 2 X

*

0.4

1 2 X

–0.41.6–1.6 ˆ 

1 2 X

1 2 X

*

))125/)(100(2cos(][   nAnx

16

2/4/2018 ECE2026, Spring 2018 31

FOLDING DIAGRAM

EXAMPLE:

y(t) has 1500 Hz component

SAMPLING FREQ = 2000 Hz

WHAT is the “FOLDED” ALIAS ?

1500 1000 500  

2/4/2018 ECE2026, Spring 2018 32

Aliasing Demo with Chirp (Lab5)

fs=8000 Hz

tt=0:1/8000:4;

xx=cos(2*pi*1000*tt.*(1+tt));

plotspec(xx+j*1e-9,8000)

grid on, shg

soundsc(xx,8000)

17

2/4/2018 ECE2026, Spring 2018 33

SAMPLING DEMO (Chap. 4)

2/4/2018 ECE2026, Spring 2018 34

ALIASING DERIVATION

 Other Frequencies give the same ˆ 

and we substitute: tnf s

If x(t)  Acos(2( f  l fs)t )

then: x[n] Acos(2( f  l fs)

n fs  )

or, x[n] Acos(2

f

fs n 2l n)

18

2/4/2018 ECE2026, Spring 2018 35

ALIASING DERIVATION–2

 Other Frequencies give the same

ˆ  Ts  2f

fs 2l

ˆ 

then : ˆ  

2( f  l fs)

fs

2 f

fs

2 l fs fs

and we want : x[n] Acos( ˆ n  )

If x(t)  Acos(2( f  l fs)t ) t n

fs

2/4/2018 ECE2026, Spring 2018 36

FOLDING DERIVATION

 Negative Freqs can give the same

x(t)  Acos(2( f  l fs)t  ) ˆ 

SAME DIGITAL SIGNAL x[n] Acos( ˆ n ) x[n] Acos((2 fTs)n 2l n  ) x[n] Acos((2 fTs)n  (2l fsTs)n ) x[n] x(nTs)  Acos(2( f  l fs)nTs  )

19

2/4/2018 ECE2026, Spring 2018 37

FOLDING (a type of ALIASING)

 MANY x(t) give IDENTICAL x[n]

 CAN’T TELL fo FROM (fs-fo)

 Or, (2fs-fo ) or, (3fs-fo )

 EXAMPLE:

 y(t) has 1000 Hz component

 SAMPLING FREQ = 1500 Hz

 WHAT is the “FOLDED” ALIAS ?

1000 1500 500

2/4/2018 ECE2026, Spring 2018 38

DIGITAL FREQ AGAIN

ˆ  Ts  2f

fs 2l

ˆ Ts   2f

fs 2l FOLDED ALIAS

ˆ 

ALIASING