Digital Signal Processing Lecture # 2 Discrete-Time ...hayes/courses/dsp/lectures/lect02.pdfDigital...

Digital Signal ProcessingLecture # 2

Discrete-Time Signals and Systems

Monson H. [email protected]

x(n) =∞∑

k=−∞

x(k)δ(n − k)

This material is the property of the author and is for the sole and exclusive use of his students. It is not for publication,nor is it to be sold, reproduced, or generally distributed.

M.Hayes (CAU-GT) Lecture # 2 September 1, 2011 1 / 37

What is Digital Signal Processing?

First, let’s ask ourselves: What is a signal?

In the fields of communications, signal processing, and in electricalengineering more generally, a signal is any time-varying orspatial-varying quantity.

I Current, voltage, acoustic waveform, light intensity across the focalplane of a camera, ...

In the physical world, any quantity measurable through time or overspace can be taken as a signal.

I Temperature distribution over an area or a volume or as a function oftime, wide velocity, rainfall, ...

Within a complex society, any set of human information or machinedata can be taken as a signal.

I Stock market prices, number of bytes transmitted across a channelversus time, human population as a function of time, ...


What is a Digital Signal?

A signal that is a function of a discrete variable, such as x(n) or z(k),is a discrete-time signal

When a discrete-time signal is quantized in amplitude,xq(n) = Q{x(n)} then we have a digital signal.


What is Digital Signal Processing?

Digital Signal Processing

DSP is concerned with the representation of signals as sequences ofnumbers and with methods of processing those numbers to achieve somedesired goal.


Examples of Processing Tasks

Information extraction1 Trends in data (forcasting and control)2 Target detection (radar and sonar)3 Single speaker in a crowd (surveillance)

Signal restoration and enhancement1 Deconvolution (deblurring, channel equalization)2 Noise reduction (CD’s?)

Signal representation1 Signal modeling (speech synthesis)2 Signal coding (Transmission and storage)3 Signal restoration, Spectral analysis


Digital Signal Processing Applications

General-Purpose DSP Graphics/Imaging Instrumentation

Digital filtering 3-D rotation Spectrum AnalysisConvolution Robot vision Function generationCorrelation Image transmission Pattern matchingHilbert transforms Image compression Seismic processingFast Fourier transforms Pattern recognition Transient analysisAdaptive filtering Image enhancement Digital filteringWindowing Homomorphic processing Phase-locked loopsWaveform generation Workstations

Voice/Speech Control Military

Voice mail Disk control Secure communicationsSpeech vocoding Servo control Radar processingSpeech recognition Robot control Sonar processingSpeaker verification Engine control Image processingSpeech enhancement Laser printer control NavigationSpeech synthesis Motor control Missile guidanceText to speech Radio frequency modems


Digital Signal Processing Applications

Telecommunications Automotive

Echo cancellation FAX Engine controlADPCM transcoders Cellular telephone Vibration analysisDigital PBX’s Speaker phones Antiskid brakesLine repeaters Digital speech interpolation Adaptive ride controlChannel multiplexing X.25 packet switching Global positioning1200 to 19200 bps modems Video conferencing NavigationAdaptive equalizers Spread spectrum Voice commandsDTMF encoding/decoding Workstations Digital radioData encryption Cellular telephones

Consumer Industrial Medical

Radar detectors Robotics Hearing aidsPower tools Numeric control Patient monitoringDigital audio/tv Security access Ultrasound equipmentMusic synthesizer Power line monitors Diagnostic toolsEducational toys Prosthetics

Fetal monitorsM.Hayes (CAU-GT) Lecture # 2 September 1, 2011 7 / 37

Objectives

Our approach, for the most part, concerns the description, analysis,and design of digital systems.

-x(n) Digital

System-y(n)

We will look at1 Types of digital systems.

F Characterizations and properties

2 Analysis of digital systems.F Finding responses to given inputs

3 Design of digital systems.F Algorithms, filters, computer programs


Digital Signal Processing Systems

-x(n) Digital

System-y(n)

Why Digital Signal Processing?

1 High precision, repeatability2 Design may be simulated in software3 Easy to modify an algorithm or system design4 Many operations difficult to implement in analog form are

straightforward using digital techniques


Discrete-Time Signals

A discrete-time signal is an indexed set of real or complex numbers:

x = {x(n)} −∞ < n <∞

I x(n) is the nth sample of the sequenceI x(n) is not defined for n not an integer.

-

6s s ss s s s s s

ss s s s s

s sq q q q q qx(n)

n

Sequences may be thought of as numbers stored in a computer arrayor as a list of numbers in a table.


Discrete-Time Signals (cont)

Where do these signals come from?

I Sampling a continuous-time signal (speech, image, audio, radar, sonar,. . .)

-

xa(t)

A/D -

x(n) = xa(nTs)

I Signals may be inherently discrete-time1 Amortization (signal is monthly debt)2 Population3 Inventory4 Stock prices


Discrete-Time Signals

Impulse sequence

δ(n) =

{1 n = 00 n 6= 0

Unit step response

u(n) =

{0 n < 01 n ≥ 0

Exponential sequencex(n) = αn

Complex sinusoidal sequencex(n) = e jω0n


Discrete-Time Systems

-

x(n)Discrete-

TimeSystem

-

y(n) = T{x(n)}

Numerical algorithm for transforming an input sequence into anoutput sequence.

Delay y(n) = x(n − nd)

Modulator y(n) = x(n) cos(nω0)

Squarer y(n) = [x(n)]2

Compressor y(n) = x(Mn) (also called a downsampler).

Smoother y(n) = 13{x(n) + x(n − 1) + x(n + 1)}


Basic Signal Manipulations

Summation: y(n) = x1(n) + x2(n)

-x1(n)

-y(n)

6

x2(n)

v

I Pointwise addition of signals


Basic Signal Manipulations (cont.)

Multiplication: y(n) = x1(n)x2(n)

-x1(n)

-y(n)

6

x2(n)

×m

I Pointwise multiplication of signals



Delay: y(n) = x(n − n0)

-x(n) z−n0 x(n − n0)

I Time shift of the sequence

-

6

s ss s s s s s

s s s

x(n)

n-

6

s s s ss s s s s s

s

x(n − 2)

n



Shift and time reversal: y(n) = x(n0 − n)

I Basic operation in convolution

-

6

s s s s s s s ss s s

x(n)

n-

6

s s s ss s s s s s s

x(7− n)

n


Signal Decomposition

An arbitrary signal may always be decomposed into a summation ofweighted and delayed unit samples.

-

6

c c c c c c c c c c c c c c c cq q q q q qx(n)

n

cx(0)

x(n) = x(0)δ(n)+x(1)δ(n − 1) + x(2)δ(n − 2) · · ·




-

6

c c c c c c c c c c c c c c cq q q q q qx(n)

n

c cx(1)

x(n) = x(0)δ(n) + x(1)δ(n − 1)+x(2)δ(n − 2) · · ·




-

6

c c c c c c c c c c c c c cq q q q q qx(n)

n

c ccx(2)

x(n) = x(0)δ(n) + x(1)δ(n − 1) + x(2)δ(n − 2)




-

6

c cc

c c c c cc

cc c c c

cc cq q q q q q

x(n)

n

c c

x(n) =∞∑

k=−∞x(k)δ(n − k)


Discrete-Time Systems

Discrete-time systems transform one sequence, the input, intoanother, the output.

-x(n)

T -y(n)

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

Generally, we would like to be able to

1 Take a given system and evaluate the response to a given input or todetermine the input that will yield a desired response.

2 Design a system to have given input/output characteristics.


Types of System Specifications

Input/Output table (catalog)

x1(n) −→ y1(n)

x2(n) −→ y2(n)

x3(n) −→ y3(n)...

Mathematical rule

(a) y(n) = [x(n)]2

(b) y(n) = 12y(n − 1) + x(n) + x(n − 1)

System properties and some input/output pairs1 T [δ(n − k)] = nu(n − k)2 Linearity


Difference Equations

An important class of LSI systems are those for which the input andoutput satisfy a linear constant coefficient difference equation:

I General Nth-order LCCDE:

N∑k=0

a(k)y(n − k) =M∑k=0

b(k)x(n − k)

The constants a(k) and b(k) specify the system.

The output may be computed recursively (assuming a(0) = 1) asfollows

y(n) =M∑k=0

b(k)x(n − k)−N∑

k=1

a(k)y(n − k)

LCCDE’s are the discrete-time counterpart of linear constantcoefficient differential equations for continuous-time systems.


Some Simple Systems

-x(n)

System -y(n)

Differentiator (First Backward Difference):

y(n) = x(n)− x(n − 1)

-

6

s ss s s s s s

s s s

x(n)

n-

6

s ss

s s s s ss

s s

y(n)

n


Some Simple Systems (cont.)

Smoother (3-point running average):

y(n) = 13 {x(n) + x(n − 1) + x(n + 1)}

-

6

s ss s s s s s

s s s

x(n)

n-

6

s s s s s s s s s s s

y(n)

n



Modulator:y(n) = x(n) cos(nω0)

-

6

s ss s s s s s

s s s

x(n)

n-

6

s ss s

ss s s

s s s

y(n)

n



Savings account:

y(n) = y(n − 1) + αy(n − 1) + x(n)

I y(n) = account balance at end of nth day.I x(n) = deposit made on the nth day.I α = 1

365p

100 = interest rate.

-

6

s s s s s s s s s s s

x(n)

n-

6

s s s s s s s s s s sy(n)

n


System Properties

-x(n)

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

Linear System: A system is linear if

T [ax1(n) + bx2(n)] = ay1(n) + by2(n)

for any complex numbers a and b and for any signals x1(n) and x2(n).


Examples

Linear Nonlinear

1. y(n) = 3x(n) 1. y(n) = x2(n)

2. y(n) = nx(n) 2. y(n) = ax(n) + b

3. y(n) = x(n) + x(−n) 3. y(n) = log x(n)


Example

If a linear system is characterized by the following input/output pairs

T [δ(n)] = u(n)

T [δ(n − 1)] = 2u(n − 1)

T [δ(n − 2)] = 3u(n − 2)...

T [δ(n − n0)] = (n0 + 1)u(n − n0)

then the response to x(n) = δ(n) + δ(n − 1)− δ(n − 2) is

y(n) = u(n) + 2u(n − 1)− 3u(n − 2) = δ(n) + 3δ(n − 1)


System Properties (cont.)

Shift-Invariance: A system is shift-invariant if a shift in the inputresults in a shift in the output

T [x(n − n0)] = y(n − n0)

for any integer n0 and for any signal x(n).

Interpretation: The system does not change in time.


Examples

Shift-Invariant Shift-Varying

1. y(n) = x(n) + x(n − 1) 1. y(n) = nx(n)

2. y(n) = x2(n) 2. y(n) = x(−n)

3. y(n) = log x(n) 3. y(n) = x(n) cos nω0



Stability (BIBO): A system is said to be stable if any bounded inputproduces a bounded output.

|x(n)| < M <∞ =⇒ |y(n)| < N <∞

Stable Unstable

1. y(n) = x2(n) 1. Savings account

2. y(n) = x(n)− x(n − 1) 2. y(n) = nx(n)

3. y(n) = ex(n) 3. y(n) =∑n

k=0 x(k)



Causality: A system is said to be causal if for any input x(n) and forany n0 the output y(n0) depends only on values of x(n) for n ≤ n0.

Interpretation: The system cannot respond prior to a stimulus.

Causal Noncausal

1. y(n) = x2(n) 1. y(n) = x(−n)

2. y(n) = x(n) + x(n − 1) 2. y(n) = x(n) + x(n + 1)

3. savings account 3. y(n) =∑n

k=0 x(k)



Invertibility: A system is said to be invertible if distinct inputsproduce distinct outputs, i.e., with

y1(n) = T [x1(n)]

y2(n) = T [x2(n)]

thenx1(n) 6= x2(n) =⇒ y1(n) 6= y2(n)

For invertible systems: The input is uniquely determined by theoutput. This is important in the deconvolution problem.


Examples

Invertible Noninvertible

1. y(n) = 2x(n) 1. y(n) = x2(n)

2. y(n) =∑n

k=−∞ x(k) 2. y(n) = nx(n)

3. y(n) = x(−n) 3. y(n) = x(n)− x(n − 1)


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