Digital Signal Processing Dr. Ahmad Salman

BEE2-CD SEECS

Quote of the Day

Mathematics is the tool specially suited for dealing with abstract concepts of any kind and

there is no limit to its power in this field.

Paul Dirac

Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall Inc.

Course Outline

• Objective :Develop basic understanding of Digital Signal

Processing Theory and Applications

– A natural extension of Signals and Systems

– Know the fundamental signal processing tools

– Know how and where to use which tool

– Develop a mathematical foundation for

advanced signal processing techniques

Course Outline

Required Text

Discrete-Time Signal Processing, 3rd Edition

Prentice Hall, Alan Oppenheim, Ronald Schafer

Lecture Notes

References

• Schaum’s Outlines: Digital Signal Processing

M. H. Hayes (Practice Problems)

• Understanding DSP-Lyons (2nd Edition)

• DSP-System analysis and design

Paulo S. R. Diniz et al.

MATLAB Exercises

• Signal Processing First J. H. McClellan et al.

• Digital Signal Processing-A computer based approach –Mitra

• Unlimited resources on internet

DSP Where !

• Communication – Signal Processing for Communication

– Equalization, Channel Estimation, Echo cancellation,

Detection, etc.

– Cellular phones, Satellite receivers, Modems, etc.

Courtesy : Lake of Soft

DSP Where !

• Speech (sound) applications

– Speech compression, speaker identification, speech

enhancement in noisy environment, special effects

– Cell Phones, MP3 Players, Movies, Dictation, Text-to-

speech,…

Speech enhancement

Speaker Recognition

DSP Where !

• Image Processing

– Image is after all a 2-D signal

Raw Image Filtered Image

Object tracking

DSP Where !

• Biomedical Signal Processing

– Magnetic Resonance, Tomography,

Electrocardiogram,…

DSP Where !

• Military

– Radar, Sonar, Space photographs, remote sensing

DSP Where !

• Mechanical

– Motor control, process control, oil and mineral

prospecting,…

• Automotive

– ABS, GPS, Active Noise Cancellation, Cruise Control,

Parking,…

Course Outline

Review (Chapter 2)

– Discrete-Time Signals and System • Discrete-Time Signals: Sequences

• Discrete-Time Systems

– Linear Time-Invariant Systems • Properties of Linear Time-Invariant Systems

• Linear Constant-Coefficient Difference Equations

– Freq. Domain Representation of Discrete-Time Signals • Representation of Sequences by Fourier Transforms

• Symmetry Properties of the Fourier Transform

• Fourier Transform Theorems

Z-Transform (Chapter 3)

• Properties of the Region of Convergence of the z-Transform

• The Inverse Z-Transform

• Z-Transform Properties

Course Outline

Sampling of Continuous-Time Signals (Chapter 4) – Periodic (Uniform) Sampling / Frequency-Domain

Representation – Reconstruction of a Bandlimited Signal from Its Samples – Discrete time processing of continuous time signals – Continuous time processing of discrete time signals

Transform Analysis of Linear Time-Invariant Systems (Chapter 5)

– The Frequency Response of LTI Systems – Constant-Coefficient Difference Equations – Frequency Response for Rational System Functions – Relationship between Magnitude and Phase – All-Pass Systems / Minimum-Phase Systems

Structures for Discrete-Time Systems (Chapter 6) – Block Diagram Representation /Signal Flow Graph

Representation – Basic Structures for IIR Systems / Transposed Forms – Basic Structures for FIR Systems – Effects of Coefficient Quantization – Effects of Round-Off Noise in Digital Filters (optional)

Course Outline

Filter Design Techniques (Chapter 7)

– Design of Discrete-Time IIR Filters from Continuous-Time Filters

– Design of FIR Filters by Windowing

– Optimum Approximation of FIR Filters (time permitting)

The Discrete-Fourier Transform (Chapter 8)

– Discrete Fourier Series

– Properties of the Discrete Fourier Series

– The Fourier Transform of Periodic Signals

– Sampling the Fourier Transform

– The Discrete Fourier Transform

– Properties of the DFT

Computation of the Discrete-Fourier Transform – FFT (Chapter 9)

Course Outline

Theory : • Quizzes (6-7, unannounced) 10% • Assignments (3-4) : 5% • OHTs (1 and 2) 30-40% • Final Exam 40 - 50% Practical : • Labs : 40-50%

– Project*: 15-20% – Final Lab Exam** : 20-30%

Discrete-Time Signals: Sequences

• 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)

Basic Sequences and Operations

• Delaying (Shifting) a sequence

• Unit sample (impulse) sequence

• Unit step sequence

• Exponential sequences

]nn[x]n[y o

0n1

0n0]n[

0n1

0n0]n[u

nA]n[x

-10 -5 0 5 10 0

0.5

1

1.5

-10 -5 0 5 10 0

0.5

1

1.5

-10 -5 0 5 10 0

0.5

1

Sinusoidal Sequences

• Important class of sequences

• An exponential sequence with complex

• x[n] is a sum of weighted sinusoids

• Different from continuous-time, the discrete-time sinusoids

– Have ambiguity of 2k in frequency

– Are not necessary periodic with 2/o

ncosnx o

jj eAA and e o

nsinAjncosAnx

eAeeAAnx

o

n

o

n

njnnjnjn oo

ncosnk2cos oo

integer an is k2

N if only Nncosncoso

ooo

Demo

Sinusoidal Sequences

Discrete-Time Systems

• Discrete-Time Sequence is a mathematical operation that maps a given input sequence x[n] into an output sequence y[n]

• Example Discrete-Time Systems

– Moving (Running) Average

– Maximum

– Ideal Delay System

]}n[x{T]n[y T{.} x[n] y[n]

]3n[x]2n[x]1n[x]n[x]n[y

]2n[x],1n[x],n[xmax]n[y

]nn[x]n[y o

Memoryless System

• Memoryless System

– A system is memoryless if the output y[n] at every value of n

depends only on the input x[n] at the same value of n

• Example Memoryless Systems

– Square

– Sign

• Counter Example

– Ideal Delay System

2]n[x]n[y

]n[xsign]n[y

]nn[x]n[y o

Linear Systems

• Linear System: A system is linear if and only if

• Examples

– Ideal Delay System

(scaling) ]n[xaT]n[axT

and

y)(additivit ]n[xT]n[xT]}n[x]n[x{T 2121

]nn[x]n[y o

T{x1[n]+x2[n]} = x1[n-no]+x2[n-no]

T{x2[n]}+T x1[n]{ } = x1[n-no]+x2[n-no]

T ax[n]{ } = ax1[n-no]

aT x[n]{ } = ax1[n-no]

Time-Invariant Systems

• Time-Invariant (shift-invariant) Systems

– A time shift at the input causes corresponding time-shift at output

• Example

– Square

• Counter Example

– Compressor System

]nn[xT]nn[y]}n[x{T]n[y oo

2]n[x]n[y

2oo

2

o1

]nn[xn-ny gives output the Delay

]nn[xny is output the input the Delay

]Mn[x]n[y

oo

o1

nnMxn-ny gives output the Delay

]nMn[xny is output the input the Delay

Causal System

• Causality

– A system is causal it’s output is a function of only the current and previous samples

• Examples

– Backward Difference

• Counter Example

– Forward Difference

]n[x]1n[x]n[y

]1n[x]n[x]n[y

Stable System

• Stability (in the sense of bounded-input bounded-output BIBO)

– A system is stable if and only if every bounded input produces a bounded output

• Example

– Square

• Counter Example

– Log

yx B]n[y B]n[x

2]n[x]n[y

2x

x

B]n[y by bounded is output

B]n[x by bounded is input if

]n[xlog]n[y 10

nxlog0y 0nxfor bounded not output

B]n[x by bounded is input if even

10

x