SIGNALS AND SYSTEMS AND INTRODUCTION TO DIGITAL SIGNAL PROCESSING CSE, RU.

Dr. Md. Ekramul HamidAssociate ProfessorDepartment of Computer Science and EngineeringUniversity of Rajshahi

SIGNALS AND SYSTEMSANDINTRODUCTION TO DIGITAL SIGNAL PROCESSINGCSE, RU

Digital Signal Processing Signal Processing deals with the enhancement, extraction,

and representation of information for communication or analysis Many different fields of engineering rely upon signal processing

technology Acoustics, telephony, radio, television, seismology, and radar are

some examples Initially, signal processing systems were implemented

exclusively with analog hardware However, recent advances in high-speed digital technology have

made discrete signal processing systems more popular. Digital systems have an advantage over analog systems in that

they can process signals with an extraordinary degree of precision

Unlike the resistive and capacitive networks of analog systems, digital systems can be built numerically with the simple operations of addition and multiplication.

Digital Signal Processing is a field of numerical mathematics

that is concerned with the processing of discrete signals This area of mathematics deals with the principles that underlie

all digital systems2

Goal of DSP

3

Typical System Components

4

Applications of DSP

5

Applications of DSP: MultimediaCompression: Fast, efficient, reliable

transmission and storage of dataApplied on audio, image and video

data for transmission over the Internet, storage

Examples: CDs, DVDs, MP3, MPEG4, JPEG

Mathematical Tools: Fourier Transform, Quantization, Modulation

6

Applications of DSP: Biological signal

Examples: Brain signals (EEG) Cardiac signals (ECG) Medical images (x-ray, PET, MRI)

Goals: Detect abnormal activity (heart attack) Help physicians with diagnosis

Tools: Filtering, Fourier Transform

7

Applications of DSP: Biometrics Identifying a person using

physiological characteristicsExamples:

Fingerprint Identification Face Recognition Voice Recognition

8

Applications of DSP: Audio Signal processing

Active noise cancellation: Adaptive filtering Headphones used in cockpits

Digital Audio Effects Add special music effects such as delay,

echo, reverbAudio signal separation

Separate speech from interference

9

Main Topics to be Covered:

o Signals And Systems – (Prerequisite of DSP)

o Linear Time Invariant Systems

o Convolution

o Correlation

10

Signals and Systems Defined

A signal is any physical phenomenon which conveys information

Systems respond to signals and produce new signals

Excitation signals are applied at system inputs and response signals are produced at system outputs

11

Signal

12

Signal (Example)

13

Independent Variable• Time is often the independent

variablefor a signal. x(t) will be used to representa signal that is a function of time, t.

• A temporal signal is defined by the relationship of its amplitude (the dependent variable) to time (the independent variable).• An independent variable can be 1D

(time), 2D (space), 3D (space) or even something more complicated.

• The signal is described as a function of this variable.

• There are many types of functions that can be used to describe signals (continuous, discrete, random).

14

http://stellar.mit.edu/S/course/6/sp08/6.003/courseMaterial/topics/topic1/lectureNotes/lecture__1/lecture__1.pdf


Analog and Digital Signal

Signals can be analog or digital.Analog signals can have an infinite

number of values in a range.Digital signals can have only a

limited number of values.

15

Analog or continuous Time Signal

• Most of the signals in the physical world are CT signals, since the time scale is infinitesimally fine (e.g., voltage, pressure, temperature, velocity).

• Often, the only way we can view these signals is through a transducer, a device that converts a CT signal to an electrical signal.

• Common transducers are the ears, the eyes, the nose… but these are a little complicated.

• Simpler transducers are voltmeters, microphones, and pressure sensors.

16


Analog Signal

-5

-3

-1

1

3

5

-10 -5 0 5 10

-5

-4

-3

-2

-1

0

1

2

3

4

5

-10 -5 0 5 10

Amplitude

Phase

Frequency

f(x) = 5 cos (x)

f(x) = 5 cos (x + 3.14)

f(x) = 5 cos (3 x + 3.14) -5

-3

-1

1

3

5

-10 -5 0 5 10

17

Signal in Time Domain The Independent Variable is Time The Dependent Variable is the Amplitude Most of the Information is Hidden in the Frequency

Content

0 0.5 1-1

-0.5

0

0.5

1

0 0.5 1-1

-0.5

0

0.5

1

0 0.5 1-1

-0.5

0

0.5

1

0 0.5 1-4

-2

0

2

4

10 Hz

2 Hz

20 Hz

2 Hz +10 Hz +

20Hz

TimeTime

Time Time

Mag

nit

ud

e

Mag

nit

ud

e

Mag

nit

ud

e

Mag

nit

ud

e

18

Analog to Digital Recording Chain

ADC

Continuously varying electrical energy is an analog of the sound pressure wave.

Microphone converts acoustic to electrical energy. It’s a transducer.

ADC (Analog to Digital Converter) converts analog to digital electrical signal.Digital signal transmits binary numbers.

DAC (Digital to Analog Converter) converts digital signal in computer to analog for your headphones.

19

Instantaneous amplitudes of continuous analog signal, measured at equally spaced points in time.

A series of “snapshots”

ADC: Step1: Sampling

20

[a.k.a. “sample word length,” “bit depth”]Precision of numbers used for measurement: the more bits, the higher the resolution.

Example: 16 bit

Sampling RateHow often analog signal is measured

Sampling Resolution

[samples per second, Hz]Example: 44,100 Hz

Analog to Digital Overview

21

Nyquist Theorem:Sampling rate must be at least twice as high as the highest frequency you want to represent.

Determines the highest frequency that you can represent with a digital signal.

Capturing just the crest and trough of a sine wave will represent the wave exactly.

Sampling Rate

22

Recovery of a sampled sine wave for different sampling rates

23

Aliasing

What happens if sampling rate not high enough?

A high frequency signal

sampled at too low a rate

looks like …

… a lower frequency signal.

That’s called aliasing or foldover. An ADC has a low-pass anti-aliasing filter to prevent this.

Synthesis software can cause aliasing.24

Anti-Aliasing Filter

An anti-aliasing filter removes frequencies that are higher than half the sampling rate using what is called a low pass filter. A low pass filter lets the low frequencies “pass" and "cut" the high frequencies. Low pass filters are sometimes called high cut filters.

25

Effects of under sampling

26

Effects of required sampling

27

Common Sampling Rates

Sampling Rate Uses

44.1 kHz (44100) CD, DAT

48 kHz (48000) DAT, DV, DVD-Video

96 kHz (96000) DVD-Audio

22.05 kHz (22050) Old samplers

Most software can handle all these rates.

Which rates can represent the range of frequencies audible by (fresh) ears?

28

A/D: Step2: Quantization

Sampling results in a series of pulses of varying amplitude values ranging between two limits: a min and a max.

The amplitude values are infinite between the two limits.

We need to map the infinite amplitude values onto a finite set of known values.

This is achieved by dividing the distance between min and max into L zones, each of height

= (max - min)/L

29

Quantization Level

The midpoint of each zone is assigned a value from 0 to L-1 (resulting in L values)

Each sample falling in a zone is then approximated to the value of the midpoint.

30

Assigning Codes to Zones Each zone is then assigned a binary code. The number of bits required to encode the

zones, or the number of bits per sample as it is commonly referred to, is obtained as follows:

nb = log2 L Given our example, nb = 3 The 8 zone (or level) codes are therefore:

000, 001, 010, 011, 100, 101, 110, and 111

Assigning codes to zones: 000 will refer to zone -20 to -15 001 to zone -15 to -10, etc. 31

0

1

2

3

4

5

6

7

Am

plit

ud

e

Time — measure amp. at each tick of sample clock

A 3-bit binary (base 2) number has 23 = 8 values.

3-bit Quantization

32

4-bit Quantization

A 4-bit binary number has 24 = 16 values.

0

2

4

6

8

10

12

14

Am

plit

ud

e

A better approximation

Time — measure amp. at each tick of sample clock

33

Quantization Error

When a signal is quantized, we introduce an error - the coded signal is an approximation of the actual amplitude value.

The difference between actual and coded value (midpoint) is referred to as the quantization error.

The more zones, the smaller which results in smaller errors.

BUT, the more zones the more bits required to encode the samples -> higher bit rate

34

Quantization Noise

Round-off error: difference between actual signal and quantization to integer values…

Random errors: sounds like low-amplitude noise

35

A/D: step3: Coding

QuantizationQuantization is the process of converting the sampled analog voltages into digital words.

Data codingData coding separates the digital words so that they are more easily identified.

36

Digital to Analog Conversion: Sample and Hold

To reconstruct analog signal, hold each sample value for one clock tick; convert it to steady voltage.

0

1

2

3

4

5

6

7

Am

plit

ud

e

Time 37

DAC: Smoothing Filter

Apply an analog low-pass filter to the output of the sample-and-hold unit: averages “stair steps” into a smooth curve.

0

1

2

3

4

5

6

7

Am

plit

ud

e

Time 38

Discrete-Time Signal (Example) Discrete-time signals are represented by sequence of

numbers The nth number in the sequence is represented with

x[n] Often times sequences are obtained by sampling of

continuous-time signals In this case x[n] is value of the analog signal at xc(nT) Where T is the sampling period

0 20 40 60 80 100-10

0

10

t (ms)

0 10 20 30 40 50-10

0

10

n (samples)39

Signals With Symmetry: Periodic

The periodicity of sequence

then the is called a periodic sequence,and the value of N is called the fundamental period.

)(nx

)()( kNnxnx if: any integerk: positive integerN

40


0 10 20 30 40 50 60 70 80 90-1

-0.5

0

0.5

1

n

Am

plit

ud

e

periodic sequence

0 10 20 30 40 50 60 70 80 90-1

-0.5

0

0.5

1

n

Am

plit

ud

e

periodic sequence

periodic sequence)

8sin( n

)16

sin( n

Signals With Symmetry: Even/Odd

)()( nxnx

)()( nxnx

Even

Odd

42


Signals With Symmetry: Even/Odd

• Any signals can be expressed as a sum of even and odd signals.

That is:

• This is demonstrated to the right for a signal referred to as

a unit step.

2/)]()([)(

2/)]()([)(

:

)()()(

txtxtx

txtxtx

where

txtxtx

odd

even

oddeven

43


The Discrete-Time Signal: Sequences

Unit sample sequence

0,0

0,1)(

n

nn

0

00 ,0

,1)(

nn

nnnn

0,0

0,1)(

n

nnu

0

00 ,0

,1)(

nn

nnnnu

)1()()( nunun

0

)()(m

mnnu

Unit step sequence

Exponential sequence nA]n[x

-10 -5 0 5 100

0.5

1

1.5

-10 -5 0 5 100

0.5

1

1.5

-10 -5 0 5 100

0.5

1

46


-10 -5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plitu

de

unit sample sequence

-10 -5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plitu

de

)(n

)5( n

47


-5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plitu

de

unit step sequence

-5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plitu

de

)(nu

)5( nu

48


Operations on sequence

Time-shifting operation

)()( Nnxny where is an integerN

delaying operation

0N

advance operation

0N

z-1)(nx )1()( nxnyUnit delay

z)(nx )1()( nxnyUnit advance

49


-10 -5 0 5 10 15 20 25 300

0.1

0.2

n

Am

plitu

de

original sequence

-10 -5 0 5 10 15 20 25 300

0.1

0.2

n

Am

plitu

de

delayed sequence

-10 -5 0 5 10 15 20 25 300

0.1

0.2

n

Am

plitu

de

advanced sequence


)(8.02.0 nun

)5(8.02.0 5 nun

)5(8.02.0 5 nun


50


Time-reversal (folding) operation

)()( nxny

Addition operation

)(nx )()()( nwnxny Adder

)(nw

Sample-by-sample addition )()()( nwnxny

51


-20 -15 -10 -5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plitu

de

original sequence

-20 -15 -10 -5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plitu

de

folding sequence

folding operation )(8.0 nun

)(8.0 nun

folding operation

52

0 5 10 15 20 25 30 35 400

0.5

1

n

Am

plit

ud

e

x1(n)

0 5 10 15 20 25 30 35 40-1

0

1

n

Am

plit

ud

e

x2(n)

0 5 10 15 20 25 30 35 40-1

0

1

2

n

Am

plit

ud

e

x1(n)+x2(n)

addition operation )(8.0 nun

)()2.0cos( nun

)()2.0cos()(8.0 nunnun


Scaling operation

)()( nAxny

)(nx )()( nAxny Multiplier A

Product (modulation) operation

)(nx )()()( nwnxny modulator

)(nw

Sample-by-sample multiplication )()()( nwnxny

54

0 20 40 60 80 100 120 140 160-0.1

0

0.1A

mp

litu

de

x1(n)

0 20 40 60 80 100 120 140 160-1

0

1

Am

plit

ud

e

x2(n)

0 20 40 60 80 100 120 140 160-0.1

0

0.1

Am

plit

ud

e

x1(n)*x2(n)

modulation operation n0125.0sin1.0

n125.0sin

)()( 21 nxnx

Decimation and Interpolation

Decimation---down-sampling

N

)()( mNxmy

x(n)

56


N

)()( mNxmy


57


N

)()( mNxmy

y(m)


58


N

Interpolation --- up-sampling

59


N

Interpolation --- up-sampling

60


Sinusoidal sequence

nnAnx ),cos()( 0

amplitude

digital angular frequency

phase

A

0

) 205.0sin(5.1)(

) 205.0cos(5.1)(

2

1

nnx

nnx

61


0 10 20 30 40 50 60-2

-1

0

1

2

n

Am

plitu

de

Sinusoidal sequence

0 10 20 30 40 50 60-2

-1

0

1

2

n

Am

plitu

de

) 205.0cos(5.1 n

) 205.0sin(5.1 n

62


The periodicity of sinusoidal sequence

)cos()( 0 nAnx

)cos()( 00 NnANnx

k

NorkN

00

22

If , : any

integerN k

is a periodic sequence and its period is)(nx

0

2

k

N 0

min2N

63


0min

2

N0

2

If is a integer

0

2

If is a noninteger rational number

QNkP

QkN

P

Q min

00

22

0

2

If is a irrational number

is an aperiodic sequence)(nx

)cos()( 0 nAnx

64

0 10 20 30 40 50 60 70 80 90-1

0

1

Am

plit

ud

e

x1(n)

0 10 20 30 40 50 60 70 80 90-1

0

1

Am

plit

ud

e

x2(n)

0 10 20 30 40 50 60 70 80 90-1

0

1

Am

plit

ud

e

x3(n)

Periodicity of sequence )8

sin( n

)103sin( n

)4.0sin( n

The Discrete-Time Signal: Sequencesns

Complex-valued exponential sequence

nenx nj ,)( )( 0

)()(

sincos)( 00

njxnx

njenenx

imre

nn

Attenuation factor

njenx

)85

1(2)(

66

0 5 10 15 20 25 30 35-0.5

0

0.5

1

1.5

2

n

Am

plit

ud

e

real part

0 5 10 15 20 25 30 35-0.5

0

0.5

1

1.5

n

Am

plit

ud

e

imaginary part

Complex-valued exponential sequence

nje

)85

1(2

The Discrete-Time System

A discrete-time system processes a given input sequence x(n) to generate an output sequence y(n) with more desirable properties.Mathematically, an operation T [ • ] is used.

y(n) = T [ x(n) ]

x(n): excitation, input signal

y(n): response, output signal

Introduction

68

Example: Accumulator

The input-output relation can also be written in the form:

This form is used for a causal input sequence, in which case y(-1) is called the initial condition

)()1()()()()(1

nxnynxlxlxnyn

l

n

l

0 ,)()1()()()(00

1

nlxylxlxny

n

l

n

ll

The output at time instant n is the sum of the input sample at time instant n and the previous output at time instant n-1, which is the sum of all previous input sample values from to n-1

)(ny)(nx

)1( ny


Classification

Linear System

Time-Invariant (Shift-Invariant) System

Linear Time-Invariant (LTI) System

Causal System

Stable System

Memory System

70


Linear System

A system is called linear if it has two mathematical properties: homogeneity and additivity.

)]([)]([)]()([ 2121 nxTnxTnxnxT

)]([)]([ nxaTnaxT

)]([)]([)]()([ 22112211 nxTanxTanxanxaT

Accumulator

71


Time-Invariant (Shift-Invariant) System

)()]([ then

)()]([ if

00 nnynnxT

nynxT

Accumulator

72


Linear Time-Invariant (LTI) System

A system satisfying both the linearity and the time-invariance properties is called an LTI system.

LTI systems are mathematically easy to analyze and characterize, and consequently, easy to design.

73


The output of an LTI system is called

linear convolution sum

)(*)()()()]([)( nhnxknhkxnxLTInyk

An LTI system is completely characterized in the time domain by the impulse response h(n).

74

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

Am

plit

ud

e

x(n)

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

Am

plit

ud

e

h(n)

0 5 10 15 20 25 30 35 40 45 500

5

10

Am

plit

ud

e

y(n)

return

)(9.0)( nunh n

)()( 10 nRnx

)()()( nhnxny


Causal System

For a causal system, changes in output samples do not precede changes in the input samples.

In a causal system, the -th output sample

depends only on input samples for

and does not depend on input samples for

0n)(nx 0nn

0nn

)2()1()()( 321 nxanxanxanye.g.

76


An LTI system will be a causal system if and only if :

0,0)( nnh

An ideal low-pass filter is not a causal system !

0,0)( nnx

A sequence is called a causal sequence if :

)(nx

77


Stable System

A system is said to be bounded-input bounded-

output (BIBO) stable if every bounded input

produces a bounded output, i.e.

PnyMnx )( then ,)( if

An LTI system will be a stable system if and only if :

n

nhS )(

78


Memory: A system is memoryless if y[n] = f ( x[n]

)▪ i.e. it sees only present values.

A system has memory if y [n] depends on previous values ▪ it can also depend on present and future

values!

79

Consider the DT SISO system:

If the input signal is and the system has no energy at , the output is called the impulse response of the system

DT Unit-Impulse ResponseDT Unit-Impulse Response

[ ]y n[ ]x n

[ ]h n[ ]n

[ ] [ ]x n n[ ] [ ]y n h n

System

System

0n

80

Consider the DT system described by

Its impulse response can be found to be

ExampleExample

[ ] [ 1] [ ]y n ay n bx n

( ) , 0,1,2,[ ]

0, 1, 2, 3,

na b nh n

n

81

Linear Time-Invariant Systems and Convolution

82


83


84


85


86


87

Correlation

Correlation addresses the question: “to what degree is signal A similar to signal B.”

An intuitive answer can be developed by comparing deterministic signals with stochastic signals. Deterministic = a predictable signal equivalent

to that produced by a mathematical function Stochastic = an unpredictable signal equivalent

to that produced by a random process

88

Correlation

Correlation is maximum when Two signals are similar in shape And are in phase (or unshifted)

Correlation is measure of similarity between two signals as a function of time shift between them

89

Correlation functions shows how similar two signals are, and how long they remain similar when one is shifted with respect to the other

Correlation

90

Autocorrelation

Correlating a signal with itself

91

Autocorrelation

Autocorrelation can be used to extract a signal from noise

92

Autocorrelation to locate a signal

Cross correlation can be used to detect and locate known reference signal in noise

93

Xcorrelation to identify a signal

Cross correlation can be used to identify a signal by comparison with a library of known reference signal

94

Convolution and Correlation

R[n]= x[k]y[n+k] Vs. C[n] = x[k]y[n-k]

95

Convolution and Correlation

If one signal is symmetric, convolution and correlation are identical

96

Thanks for your attention

97

SIGNALS AND SYSTEMS AND INTRODUCTION TO DIGITAL SIGNAL PROCESSING CSE, RU.

Documents

digital signals

analog signals

o signals

digital systems

processing of discrete

ct signals

new signals excitation

response signals