Chapter1

COURSE:COURSE:DIGITAL SIGNAL TRANSFORMS & DIGITAL SIGNAL TRANSFORMS &

APPLICATIONSAPPLICATIONS

Instructor: Hoang Le Uyen Thuc

Electronic and Telecommunication Engineering Department

Danang University of Technology

[email protected]

Where are you now? Where are you now? Where will you go to?Where will you go to?

Digital Signal Transforms & Applications

DSP FINAL PROJECT

CT Signals & Systems

You are here

GoalsGoals

1. To provide students the basic digital signal

transforms such as Z-transform, Discrete-Time

Fourier Transform and Discrete Fourier Transform

and their applications in analysis and synthesis of

DSP systems

2. To equip students the basic 2D transforms using in

image processing

GradingGrading

1. Homework: 30%

Submit right before the examination

2. Exam: 70%

Things can be brought into the exam. room: pen/pencils,

ruler, eraser, clock, calculator, white drafts, two A4

papers

Each paper is two-side personal writing

Exam. duration: 60 minutes

Textbooks and ReferencesTextbooks and References

Textbook: John G.Proakis & Dimitris G. Manolakis - Digital signal

processing - Prentice Hall, New Jersey 2006

References:

[1] Nguyễn Quốc Trung - Xử lý tín hiệu & lọc số Tập 1- NXB Khoa

học & kỹ thuật, Hà Nội 2001

[2] Joyce Van de Vegte - Fundamentals of Digital Signal

Processing - Prentice Hall 2002

[3] Vinay K.Ingle & John G.Proakis - Digital Signal Processing

using Matlab - Book Ware Companion Series 2007

Schedule:Schedule:

Chapter 1: Digital Signals and Systems (5 hrs)

Chapter 2: Z-transforms and applications (5 hrs)

Chapter 3: Discrete-time Fourier Transform and applications (5

hrs)

Chapter 4: Discrete Fourier Transform and applications (5 hrs)

Chapter 5: 2D Transforms and application in image processing (5

hrs)

Review and class discussion (5 hrs)

CHAPTER 1:CHAPTER 1:

DIGITAL SIGNALS & SYSTEMSDIGITAL SIGNALS & SYSTEMS

Lesson #1: A big picture about Digital Signal Processing

Lesson #2: Digital signals

Lesson #3: Digital systems

Duration: 5 hrs

Lecture #1:Lecture #1: A big picture about A big picture about

Digital Signal Processing Digital Signal Processing

Duration: 1 hr

Outline:

1. Signals

2. Digital Signal Processing (DSP)

3. Why DSP?

4. DSP applications

SignalsSignals

Function of independent variables such as time, distance,

position, temperature

Convey information

Examples:

1D signal: speech, music, biosensor…

2D signal: image

2.5D signal: video (2D image + time)

3D signal: animated

11--D signalsD signals

Color imageSpeech signal

ECG

EEG

22--D image signalsD image signals

Binary image Color imageGrey image

2.52.5--D video signalsD video signals

33--D animated signalsD animated signals

What is Digital Signal Processing?What is Digital Signal Processing?

Represent a signal by a sequence of numbers (called a

"discrete-time signal” or "digital signal").

Modify this sequence of numbers by a computing process

to change or extract information from the original signal

The "computing process" is a system that converts one

digital signal into another— it is a "discrete-time system” or

"digital system“.

Transforms are tools using in computing process

DiscreteDiscrete--time signal vs. time signal vs. continuouscontinuous--time signaltime signal

Continuous-time signal:

- define for a continuous duration of time

- sound, voice…

Discrete-time signal:

- define only for discrete points in time (hourly, every second, …)

- an image in computer, a MP3 music file

- amplitude could be discrete or continuous

- if the amplitude is also discrete, the signal is digital.

Analog signal vs. digital signalAnalog signal vs. digital signal

Analog signal Digital signal

00 10 00 10 11

Scheme for the Digital Signal Processing Scheme for the Digital Signal Processing of an analog signalof an analog signal

Digital Signal Processing Digital Signal Processing ImplementationImplementation

Performed by:

Special-purpose (custom) chips: application-specific integrated

circuits (ASIC)

Field-programmable gate arrays (FPGA)

General-purpose microprocessors or microcontrollers (µP/µC)

General-purpose digital signal processors (DSP processors)

DSP processors with application-specific hardware (HW)

accelerators

Digital Signal Processing Digital Signal Processing implementationimplementation

Digital Signal Processing Digital Signal Processing ImplementationImplementation

Use basic operations of addition,

multiplication and delay

Combine these operations to accomplish

processing: a discrete-time input signal

another discrete-time output signal

Two main categories of DSPTwo main categories of DSP

Analysis Filtering

Measurement Digital Signals

- feature extraction

- signal recognition

- signal modeling

........

- noise removal

- interference

removal

……

Digital Signals

Signal Analysis and FilteringSignal Analysis and Filtering

Analysis Filtering

Measurement Digital Signals

Digital Signals

Transforms:

ZT

DTFT

DFT/FFT

FIR

IIR

Adaptive filters

Advantages of Digital Advantages of Digital Signal ProcessingSignal Processing

Flexible: re-programming ability

More reliable

Smaller, lighter less power

Easy to use, to develop and test (by using the

assistant tools)

Suitable to sophisticated applications

Suitable to remote-control applications

Limitations of Digital Signal Limitations of Digital Signal ProcessingProcessing

Aliasing in sampling: taking samples at intervals and don’t

know what happens in between can’t distinguish higher

and lower frequencies avoid: sampling theory

Limitations of Digital Signal Limitations of Digital Signal ProcessingProcessing

Quantization error: due to the limited number of bit

available smoothly varying signal represented by stepped

waveform limited precision in data storage and arithmetic

DSP DSP applications - radar

DSP applications DSP applications -- biomedicalbiomedical

Analysis of biomedical signals, diagnosis, patient

monitoring, preventive health care

DSP applications DSP applications –– speech compressionspeech compression

DSP applicationsDSP applications

Speech recognition:

DSP applications DSP applications -- communicationcommunication

Digital telephony: transmission of information in

digital form via telephone lines, modern technology,

mobile phone

DSP applications DSP applications –– image processingimage processing

Image enhancement: processing an image to be more

suitable than the original image for a specific application

It makes all the difference whether one sees darkness through It makes all the difference whether one sees darkness through It makes all the difference whether one sees darkness through It makes all the difference whether one sees darkness through

the light or brightness through the shadowsthe light or brightness through the shadowsthe light or brightness through the shadowsthe light or brightness through the shadows

David LindsayDavid LindsayDavid LindsayDavid Lindsay

DSP applications DSP applications –– image processingimage processing

Image compression: reducing the redundancy in the

image data

DSP applications DSP applications –– image processingimage processing

Image restoration: reconstruct a degraded image using a

priori knowledge of the degradation phenomenon

DSP applicationsDSP applications-- musicmusic

Recording, encoding, storing

Playback

Manipulation/mixing

DSP applicationsDSP applications-- finger print recognitionfinger print recognition

Lecture #2Lecture #2

Digital (DT) SignalsDigital (DT) Signals

Duration: 2 hrs

Outline:

1. Representations of DT signals

2. Some elementary DT signals

3. Classification of DT signals

4. Simple manipulations of DT signals

Converting a CT signal into a DT signal by sampling: given xa(t) to

be a CT signal, xa(nT) is the value of xa(t) at t = nT DT signal is

defined only for n an integer

∞<<∞−≡==

n),n(x)nT(x)t(xa

nTta

-2T -T 0 T 2T 3T 4T 5T 6T 7T . . . nT

t

Sampled signalsSampled signals

n 9 -1 0 1 2 3 4 9

x[n] 9 0 0 1 4 1 0 9

1. Functional representation

≠

=

=

=

n,0

2n,4

3,1n,1

]n[x

Representations of DT signalsRepresentations of DT signals

2. Tabular representation

3. Sequence representation

Representations of DT signalsRepresentations of DT signals

4. Graphical representation

1,4,1,0][↑

=nx

-1 0 1 2 3 4 5 n

4

1 1

1. Unit step sequence

2. Unit impulse signal

3. Sinusoidal signal

4. Exponential signal

Some elementary DT signalsSome elementary DT signals

1 0[ ]

0 0

nu n

n

, ≥=

, <

-1 0 1 2 3 4 5 6 n

1 1 1

Unit stepUnit step

-n0-1 n0 n0+1 n

1 1 1

TimeTime--shifted unit stepshifted unit step

<

≥=−

0

0

0nn,0

nn,1]nn[u

Unit impulseUnit impulse

1 0[ ]

0 0

nn

nδ

, ==

, ≠

-2 -1 0 1 2 3 n

1

TimeTime--shifted unit impulseshifted unit impulse

n0-1 n0 n0+1 n

1

≠

==−δ

0

0

0nn,0

nn,1]nn[

Relation between unit step and Relation between unit step and unit impulseunit impulse

]n[x]nn[]n[x

]nn[]n[x]nn[]n[x

]1n[u]n[u]n[

]k[]n[u

0

n

0

000

n

k

=−δ

−δ=−δ

−−=δ

δ=

∑

∑

∞

−∞=

−∞=

Sinusoidal signalSinusoidal signal

+∞<<∞−θ+π=

+∞<<∞−θ+Ω=

n),nF2cos(A

n),ncos(A)n(x

-20 -15 -1 0 -5 0 5 1 0 15 20-1 .5

-1

-0 .5

0

0 .5

1

1 .5

Exponential signalExponential signal

nCa]n[x =

1. If C and a are real, then x[n] is a real exponential

a > 1 growing exponential

0 < a < 1 shrinking exponential

-1 < a < 0 alternate and decay

a < -1 alternate and grows

2. If C or a is complex, then x[n] is a complex exponential

Exponential signal Exponential signal -- ExampleExample

nj

enx

+−

= 612

1

2][

π

0 5 10 15 20 25 30 35 40-2

-1

0

1

2Real part

0 5 10 15 20 25 30 35 40-1

0

1

2Imaginary part

Exponential signal Exponential signal -- ExampleExample

nnx )2.1)(2.0(][ =

0 5 10 15 20 25 30 350

20

40

60

80

100

120

Tim e index n

Am

plit

ude

Periodic and aperiodic signals

Symmetric (even) and antisymmetric (odd) signals

Energy and power signals

Classification of DT signalsClassification of DT signals

Periodic signals: important, both for practical

reasons and for mathematical analysis

DT sinusoidal signal is periodic only if its frequency f

is rational number

Periodic and Aperiodic signalsPeriodic and Aperiodic signals

ExamplesExamples

6

1[ ]j n

x n eπ

=

32 5[ ] sin( 1)x n nπ= +

3[ ] cos(2 )x n n π= −

Determine which of the signals below are periodic. For the ones that are, find the fundamental period and fundamental frequency

Any DT signal can be expressed as the sum of an even signal and

an odd signal

Even and Odd signalsEven and Odd signals

Even [ ] [ ]e ex n x n: = −

Odd [ ] [ ]o ox n x n: = − −12

[ ] ( [ ] [ ])ex n x n x n= + −

12

[ ] ( [ ] [ ])ox n x n x n= − −

[ ] [ ] [ ]e ox n x n x n= +

Given x[n] = [ 1 1 2 2 0 1 2 2 ]

Find xe[n] and xo[n]

ExamplesExamples

Define the signal energy:

Define the signal power:

E is finite x[n] is called an energy signal

E is finite P = 0

E is infinite P maybe finite or infinite. If P is finite and

nonzero x[n] is called power signal

Energy and Power signalsEnergy and Power signals

∑∞

−∞=

=n

2]n[xE

∑−=

∞→ +=

N

Nn

2

N]n[x

1N2

1limP

Determine which of the signals below are energy signals?

Which are power signals?

ExamplesExamples

(a) Unit step

(b)

<

≥=

0n,)2(

0n,)2/1(]n[x

n

n

(c) ])4n[u]n[u(n4

cos]n[x −−

π=

Transformation of time:

- Time shifting

- Time scaling

- Time reversal

Adding and subtracting signals

Simple manipulations of DT Simple manipulations of DT signalssignals

x[n] x[n - k]; k is an integer

k > 0: right-shift x[n] by |k| samples

(delay of signal)

k < 0: left-shift x[n] by |k| samples

(advance of signal)

Time shifting a DT signalTime shifting a DT signal

Examples of time shiftingExamples of time shifting

-1 0 1 2 3 4 5 n

4

1 1

x[n]

-2 -1 0 1 2 n

4

1 1

0 1 2 3 4 5 n

4

1 1

x[n-2]x[n+2]

x[n] x[an]

|a| > 1: speed up by a factor of a

a must be an integer

|a| < 1: slow down by a factor of a

a = 1/K; K must be an integer

Time scaling a DT signalTime scaling a DT signal

Examples of time scalingExamples of time scaling

-1 0 1 2 3 4 5 n

4

1 1

x[n]

-1 0 1 2 n

4

-1 0 1 2 n

1

x[2n+1]x[2n]

Examples of time scalingExamples of time scaling

-1 0 1 2 3 4 5 n

4

1 1

x[n]

??13

142

??11

000

y[n]=x[n/2]x[n]n

How do we find y[1] and y[3]??

One solution is linear interpolation used in a simple compression scheme

1/2

1/2

x[n] x[-n]

Flip a signal about the vertical axis

Time reversal a DT signalTime reversal a DT signal

Examples of time reversalExamples of time reversal

-1 0 1 2 3 4 5 n

4

1 1

x[n]

-5 -4 -3 -2 -1 0 1 n

4

1 1

x[-n]

x[n] x[-n-k]

Method 1: Flip first, then shift

Method 2: Shift first, then flip

Combining time reversal and Combining time reversal and time shiftingtime shifting

Examples of combining time Examples of combining time reversal and time shiftingreversal and time shifting

-1 0 1 2 3 4 5 n

4

1 1

x[n]x[-n+1]??

Examples of combining time Examples of combining time reversal and time shiftingreversal and time shifting

-1 0 1 2 3 4 5 n

4

1 1

x[n]x[-n-1]??

x[n] x[an-b]

Be careful!!!

To make sure, plug values into the table to check

Combining time shifting and Combining time shifting and time scalingtime scaling

Examples of combining time Examples of combining time scaling and time shiftingscaling and time shifting

-1 0 1 2 3 4 5 n

4

1 1

x[n] y[n] = x[2n-3]??

004

113

142

011

000

00-1

y[n]x[n]n

-1 0 1 2 3 4 5 n

1 1y[n]

Do it “point by point”

Can do using a table, or graphically, or by

computer program

Example: x[n] = u[n] – u[n-4]

Adding and Subtracting signalsAdding and Subtracting signals

n <=-1 0 1 2 3 >=4

x[n] 0 1 1 1 1 0

Find

ExerciseExercise

[ ] ( [ 1] [ 5])( [2 ])x n u n u n nu n= + − − −

-1 0 1 2 3 4 5 n

21

-1

x[n]

Prob.1 The following graph is of signal x[n].

Plot the following:

a) 3x[-n+1]

b) x[2n] - 1

c) –x[n] + 2

d) x[2n+1]

HWHW

Prob.2 Sometimes signals can be decomposed into combinations of

simple unit sequences such as this one:

Sketch y[n] and the following signals:

a) 2-3y[n]

b) 3y[n-2]

c) 2-2y[-2+n]

HWHW

Prob.3

Decompose y[n] from Prob. 2 into its even and odd parts. Plot

these signals and show the symmetries of the plots that can be

used to visually verify their parity

HWHW

Lecture #3Lecture #3

DT systemsDT systems

Duration: 2 hrs

Outline:

1. Input-output description of systems

2. DT system properties

3. LTI systems

Think of a DT system as an operator on DT signals:

• It processes DT input signals, to produce DT output signals

• Notation: y[n] = Tx[n] y[n] is the response of the

system T to the excitation x[n]

• Systems are assumed to be a “black box” to the user

InputInput--output description of DT systemsoutput description of DT systems

Determine the response of the following system to the unit

impulse signal

ExampleExample

............................

)8/5)(4/1(]1[2

1]2[

2

1]1[

4

1]2[

8/52/14/1]0[2

1]1[

2

1]0[

4

1]1[

2/1]1[2

1]0[

2

1]1[

4

1]0[

=++=

=+=++=

=−++−=

δδ

δδ

δδ

yy

yy

yy

]1[)8/5()4/1(][)2/1(][ 1 −+= − nunny nδ

]1n[x2

1]n[x

2

1]1n[y

4

1]n[y −++−=

• Memory

• Invertibility

• Causality

• Stability

• Linearity

• Time-invariance

DT system propertiesDT system properties

y[n0] = f(x[n0]) system is memoryless

Otherwise, system has memory, meaning that its output

depends on inputs rather than just at the time of the output

Ex:

a) y[n] = x[n] + 5

b) y[n]=(n+5)x[n]

c) y[n]=x[n+5]

MemoryMemory

InvertibilityInvertibility

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

T() Ti()

x[n] x[n]

System Inverse system

Examples for Examples for invertibilityinvertibility

Determine which of the systems below are invertible

a) Unit advance y[n] = x[n+1]

b) Accumulator

c) Rectifier y[n] = |x[n]|

∑−∞=

=n

k

kxny ][][

The output of a causal system (at each time) does not

depend on future inputs

All memoryless systems are causal

All causal systems can have memory or not

CausalityCausality

Examples for causalityExamples for causality

Determine which of the systems below are causal:

a) y[n] = x[-n]

b) y[n] = (n+1)x[n-1]

c) y[n] = x[(n-1)2]

d) y[n] = cos(w0n+x[n])

e) y[n] = 0.5y[n-1] + x[n-1]

If a system “blow up” it is not stable

In particular, if a “well-behavior” signal (all values have

finite amplitude) results in infinite magnitude outputs,

the system is unstable

BIBO stability: “bounded input – bounded output” –

if you put finite signals in, you will get finite signals out

StabilityStability

Examples for stabilityExamples for stability

Determine which of the systems below are BIBO stable:

a) A unit delay system

b) An accumulator

c) y[n] = cos(x[n])

d) y[n] = ln(x[n])

e) y[n] =exp(x[n])

Scaling signals and adding them, then processing through the system

same as

Processing signals through system, then scaling and adding them

LinearityLinearity

If T(x1[n]) = y1[n] and T(x2[n]) = y2[n]

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

If you time shift the input, get the same output, but with the

same time shift

The behavior of the system doesn’t change with time

TimeTime--invarianceinvariance

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

then T(x[n-n0]) = y[n-n0]

Examples for linearity and Examples for linearity and timetime--invarianceinvariance

Determine which of the systems below are linear, wichones are time-invariant

a) [ ] [ ]y n nx n=

Examples for linearity and Examples for linearity and timetime--invarianceinvariance

Determine which of the systems below are linear, wichones are time-invariant

b) ]n[x]n[y 2=

Examples for linearity and Examples for linearity and timetime--invarianceinvariance

Determine which of the systems below are linear, wichones are time-invariant

c) ∑=

−=M

r

r rnxbny0

][][

Prob.4 Which systems are linear? Which ones are time-invariant?

HWHW

∑

∑

=

−∞=

=

=

=

=

+Ω=

=

n

k

n

k

kxnyf

kxnye

nunxnyd

nxnyc

nxnnyb

nxnya

0

0

][][)

][][)

][][][)

]2[][)

])[cos(][)

])[sin(][)

Prob.5 For the following systems:

Prove or disprove that they are:

Memoryless

Invertible

Causal

Stable

Time-invariant

Linear

HWHW

Ζ∈−=

−+=

=

Ζ∈+=

∑

∑

−

−=

−=

Mknxnyd

nxnxnyc

exnyb

akaxnya

M

k

n

n

nk

,][][)

]1[5.0][5.0][)

)(][)

,][][)

1

0

||

Method 1: based on the direct solution of the input-output equation

for the system

Method 2:

• Decompose the input signal into a sum of elementary signals

• Find the response of system

to each elementary signal

• Add those responses to obtain

the total response of the system

to the given input signal

Computing the response of DT LTI Computing the response of DT LTI systems to arbitrary inputssystems to arbitrary inputs

∑

∑

=→

→

=

k

kk

kk

k

kk

nycnynx

nynx

nxcnx

][][][

][][

][][

Convolution: an operation between the input signal to a

system and its impulse response, resulting in the output signal

In CT systems: convolution of 2 signals involves integrating

the product of the 2 signals – where one of signals is flipped

and shifted

In DT systems: convolution of 2 signals involves summing

the product of the 2 signals – where one of signals is flipped

and shifted

DT convolution formulaDT convolution formula

We can describe any DT signal x[n] as:

Example:

Impulse representation of DT Impulse representation of DT signalssignals

[ ] [ ] [ ]k

x n x k n kδ∞

=−∞

= −∑

-1 0 1 2 3 n

x[n]

-1 0 1 2 n

x[0]δ[n-0]

-1 0 1 2 n

x[1]δ[n-1]

-1 0 1 2 n

x[2]δ[n-2]+ +

Impulse response: the output results, in response to a unit

impulse

Denotation: hk[n]: impulse response of a system, to an impulse

at time k

Impulse response of DT systemsImpulse response of DT systems

Time-invariant DT system

Time-invariant DT system

δ[n]

δ[n-k]

h[n]

h[n-k]

Response of LTI DT systems to Response of LTI DT systems to arbitrary inputsarbitrary inputs

LTI DT systemδ[n-k] h[n-k]

LTI DT system

[ ] [ ] [ ]k

x n x k n kδ∞

=−∞

= −∑ ∑∞

−∞=

−=k

knhkxny ][][][

Notation: y[n] = x[n] * h[n]

Convolution sum

Commutative law

Associative law

Distributive law

Convolution sum propertiesConvolution sum properties

Commutative lawCommutative law

][*][][*][ nxnhnhnx =

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

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

Associative lawAssociative law

])[*][(*][][*])[*][( 2121 nhnhnxnhnhnx =

h1[n]x[n] y[n]

h2[n]

h2[n]x[n] y[n]

h1[n]

h1[n]*h2[n]x[n] y[n]

Distributive lawDistributive law

])[*][(])[*][(])[][(*][ 2121 nhnxnhnxnhnhnx +=+

h1[n] + h2[n]x[n] y[n]

h1[n]

x[n] y[n]

h2[n]

Computing the convolution sumComputing the convolution sum

Suppose to compute the output y[n] at time n = n0.

1. Fold h[k] about k = 0, to obtain h[-k]

2. Shift h[-k] by n0 to the right (left) if n0 is positive (negative), to obtain h[n0-k]

3. Multiply x[k] and h[n0-k] for all k, to obtain the product

x[k].h[n0-k]

4. Sum up the product for all k, to obtain y[n0]

Repeat from 2-4 fof all of n

∑∞

−∞=

−=k

knhkxny ][][][

The length of the convolution The length of the convolution sum resultsum result

Suppose:

Length of x[k] is Nx N1 ≤ k ≤ N1 + Nx – 1

Length of h[n-k] is Nh N2 ≤ n-k ≤ N2 + Nh – 1

N1 + N2 ≤ n ≤ N1 + N2 + Nx + Nh – 2

Length of y[n]:

Ny = Nx + Nh – 1

[ ] [ ] [ ] [ ] [ ]k

y n x n h n x k h n k∞

=−∞

= ∗ = −∑

Examples for computing the Examples for computing the convolution sumconvolution sum

Ex1. Find y[n] = x[n]*h[n] where

[ ] [ 1] [ 3] [ ]x n u n u n nδ= + − − + ( )[ ] 2 [ ] [ 3]h n u n u n= − −

n

n

x[n]

h[n]

-1 0 1 2 3

-1 0 1 2 3

h[-k]h[k]

Ex1 (cont.)Ex1 (cont.)

x[k]

-1 0 1 2 3 k -1 0 1 2 3 k -2 -1 0 1 k

-4 -3 -2 -1 0 k

h[2-k]

-1 0 1 2 k

h[-1-k] h[1-k]

-2 -1 0 1 2 k

y[n<-1] = 0; y[-1] = 2; y[0] = 6; y[1] = 8;

y[2] = 8; y[3] = 4; y[4] = 2; y[n>4] = 0

Examples for computing the Examples for computing the convolution sumconvolution sum

Ex2. Find y[n] = x[n]*h[n] where [ ] [ ]nx n a u n= [ ] [ ]h n u n=

1. Do it graphically

2. Use convolution formula

Examples for computing the Examples for computing the convolution sumconvolution sum

Ex3. Find y[n] = x[n]*h[n] where x[n] = bnu[n] and h[n] = anu[n+2]

|a| < 1, |b| < 1, a ≠ b

DT LTI properties based on DT LTI properties based on impulse responseimpulse response

1. Memoryless system: h[n] = Kδ[n]

2. Invertible system: h[n]*hi[n] = δ[n]

3. Causal system: h[n] is zero for all time n<0

4. BIBO stable system:

∞<∑+∞

−∞=n

nh ][

Examples of DT LTI propertiesExamples of DT LTI properties

1. What is the inverse of h[n] = 3δ[n+5]?

2. Is h[n] = 0.5nu[n] BIBO stable? Causal?

3. Is h[n] = 3nu[n] BIBO stable? Causal?

4. Is h[n] = 3nu[-n] BIBO stable? Causal?

Prob.6 Find y[n] = x[n]*h[n] where:

a) x[n] = anu[n] and h[n] = u[n] – u[n-10]

b) x[n] = u[-n] and h[n] = anu[n-2], |a|<1

c) x[n] = 2δ[n+2] + 2δ[n+1] + 2δ[n-1] + 2δ[n-2] + 2δ[n-3] + 2δ[n-4]

and h[n] = δ[n] – δ[n-1] + δ[n-2]

d) x[n] = u[-n+2] and h[n] = anu[-n]

e) x[n] = 0.2nu[n] and h[n] = δ[n] – 0.2δ[n-1]

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Prob.7 Consider the LTI system with the input and output related by: y[n] = 0.5x[n-1] + 0.7x[n]

a) Find the impulse response h[n]

b) Is this system causal? Stable? Why?

c) Determine the system response y[n] for the input shown in Fig. (a)

d) Consider the interconnections of the LTI systems given in Fig. (b). Find the impulse

response of the total system

e) Solve for the response of the system of part (d) for the input of part (c)

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Prob.8 Determine the causality and the BIBO stability for the

systems with the following impulse responses:

a) h[n] = sin(-n)u[n]

b) h[n] = e-nu[-n]

c) h[n] = enu[n]

d) h[n] = sin(n)u[-n]

e) h[n] = ne-nu[n]

f) h[n] = e-nsin(n)u[n]

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General form:

LTI systems characterized by linear LTI systems characterized by linear constant coefficient difference constant coefficient difference equationsequations

][...]1[][][...]1[][ 101 Mnxbnxbnxbnyanyany M −++−+=−++−+

1a,]rn[xb]kn[ya 0

M

0r

r

N

0k

k =−=−⇔ ∑∑==

N, M: non-negative integers

N: order of equation

ak, br: constant real coefficients

1) Put y[n] on the left hand side by itself

y[n] = -a1y[n-1] - … - aNy[n-N] + b0x[n] + … + bMx[n-M]

2) To calculate a given output at time n = n0, that is y[n0], we add

the weighted M+1 inputs b0x[n0] + … + bMx[n0-M] to the

weighted N past outputs –a1y[n0-1] - … - aNy[n0-N]

3) Increase the time index to n = n0+1 and recursively calculate the

next output. This can continue forever.

Recursive solution of Recursive solution of difference equationsdifference equations

To start this recursion somewhere, for example at n0 = 0, we need to know the N initial conditions y[n0-1], y[n0-2], …, y[n0-N]

Solve iteratively to find the 1st 3 terms of y[n] – 2y[n-1] = x[n-1]

with initial condition y[-1] = 10 and with the input x[n] = 2u[n]

Example of recursive solution Example of recursive solution the difference equationthe difference equation

Total response = zero-input component + zero-state component

= natural response + forced response

= complementary response + particular response

Closed form solutions of Closed form solutions of difference equationsdifference equations

1. Find the complementary response, assume input = 0.

2. Find the particular response, assume all initial conditions = 0. Choose the form of the particular response same as the form of input

3. Total response = complementary + particular. Use initial conditions to find N constants from the complementary response

Example for closed form Example for closed form solutionssolutions

Given y[n] – 0.3y[n-1] = x[n] with y[-1] = 0 and x[n] = (0.6)n

Example (cont)Example (cont)

Combining particular and complementary solutions:

Prob.9 Determine the response y[n] for n≥0 of the system

described by the following equation:

y[n] -0.7y[n-1] + 0.1y[n-2] = x[n] – 3x[n-1]

The input is x[n] = (-1)n and y[-2] = 29/9, y[-1] = 7/9

HWHW