COURSE NAME

DIGITAL SIGNAL PROCESSING

COURSE CODE: EC 701

Dr. Mrutyunjay Rout

Dept. of Electronics and communication Engineering

NIT Jamshedpur

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Course DescriptionUNIT-I: DSP Preliminaries, Sampling, DT signals, sampling theorem in time domain, sampling of analog

signals, recovery of analog signals, and analytical treatment with examples, mapping between analog

frequencies to digital frequency, representation of signals as vectors, concept of Basis function and

orthogonality. Basic elements of DSP and its requirements, advantages of Digital over Analog signal

processing.

UNIT-II: Discrete Fourier Transform, DTFT, Definition, Frequency domain sampling , DFT, Properties of

DFT, circular convolution, linear convolution, Computation of linear convolution using circular

convolution, FFT, decimation in time and decimation in frequency using Radix-2 FFT algorithm, Linear

filtering using overlap add and overlap save method, Introduction to Discrete Cosine Transform

UNIT-III: Z transform, Need for transform, relation between Laplace transform and Z transform, between

Fourier transform and Z transform, Properties of ROC and properties of Z transform, Relation between

pole locations and time domain behaviour, causality and stability considerations for LTI systems, Inverse

Z transform, Power series method, partial fraction expansion method, Solution of difference equations.

UNIT-IV: IIR Filter Design, Concept of analog filter design (required for digital filter design), Design of

IIR filters from analog filters, IIR filter design by approximation of derivatives filter design by impulse

invariance method, Bilinear transformation method, warping effect. Characteristics of Butterworth filters,

Chebyshev filters and elliptic filters, Butterworth filter design, IIR filter realization using direct form,

cascade form and parallel form, Finite word length effect in IIR filter design.

UNIT-V: FIR Filter Design, Ideal filter requirements, Gibbs phenomenon, windowing techniques,

characteristics and comparison of different window functions, Design of linear phase FIR filter using

windows and frequency sampling method. FIR filters realization using direct form, cascade form and

lattice form, Finite word length effect in FIR filter design, Multirate DSP, Introduction to DSP Processor

Concept of Multirate DSP, Sampling rate conversion by a non-integer factor, Design of two stage sampling

rate converter, General Architecture of DSP, Introduction to Code composer studio, Application of DSP to

Voice Processing, Music Processing, Image processing and Radar processing

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Books

Text Books:

1. John G Proakis and Manolakis, “Digital Signal Processing Principles,

Algorithms andApplications”, Pearson, Fourth Edition, 2007.

2. S.Salivahanan, A. Vallavaraj, and C. Gnanapriya, “Digital Signal

Processing”, TMH/McGraw Hill International, 2007.

Reference Books:

1. S.K. Mitra, “Digital Signal Processing, A Computer-Based Approach”,

Tata Mc Graw Hill, 1998.

2. Ifaeachor E.C, Jervis B. W., “Digital Signal processing: Practical

approach”, Pearson publication, Second edition, 2002.

3. Johny R. Johnson, Introduction to Digital Signal Processing, PHI, 2006.

Lecture: 1-8

Introduction to Digital Signal

Processing

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Lecture: 1-8➢ Signal processing emerged soon after World War I in the

form of electrical filtering.

➢ With the invention of the digital computer and the rapidprogress in VLSI technology during the 1960s, a new wayof processing signals the signal processing is term asdigital signal processing.

➢ Digital signal processors take the real world signals likeaudio, video, speech etc., that have been sampled andquantized and then mathematically manipulate them.

➢ Signals need to be processed so that the information thatthey contain can be displayed, analyzed, or converted toanother type of signal that may be of use.

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Lecture: 1-8

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What is Signal?

• Anything that carries information and represents

as a function of independent variables such as

time, space, temperature, pressure, etc.

• Any physical quantity that can be varied in such a

way as to convey information.

• A signal is any quantity that depends on one or

more independent variables.

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Example of Signal• A radio signal represents the strength of an electromagnetic wave

that depends on one independent variable, namely time is a 1-Dsignal.

• Image is a 2-D signal.

• A video signal is a 3-D signal.

• Natural signals:✓ Signals produced by the brain and heart

✓ Signals originating in galaxies, astronomical images etc.

✓ Speech signals, sounds made by dolphins

✓ Signals produced by lightning, the atmospheric pressure etc.

• Man-made signals:✓ Signals originating from satellites, radio, telephone, TV

✓ Signals due to ECG, EEG etc.

✓ signals generate from musical instruments

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• Signal Processing: Process of operation in which the characteristics of asignal such as amplitude, shape, phase, frequency, etc. undergoes a change.

OR

Signal processing is the analysis, interpretation and manipulation of anysignals like sound, images etc.

• Types of signal processing:

✓Analog Signal Processing

✓ Digital Signal Processing

Analog Signal

Processing

Analog SignalX(t)

Analog Output Signal

y(t)

Sample and

Hold

AnalogInput Signal

X(t)

Analog Output Signal

y(t)

A/D

Converter

Digital Signal

Processor

D/A

Converter

• Digital Signal processors (DSP) take real-world signals like audio, video,

pressure, temperature etc. that have been digitized and then mathematically

manipulate them

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Components of a DSP System

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• Advantages of Digital Signal Processing:✓Greater Accuracy

✓Cheaper

✓Ease of Data storage

✓Easy Operation

✓Flexibility

✓Multiplexing

• Limitations of Digital Signal Processing:✓Antialiasing Filter

✓Bandwidth limited by Sampling Rate

✓Quantization Error

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• Applications of Digital SignalProcessing:✓In Communication

✓Consumer Application (e.g., TV, FMradio etc.)

✓Image processing

✓In Biomedical

✓In Radar and Sonar

✓In Speech and Music

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• Any unwanted signal interfering with the main signal

is termed as noise. So, noise is also a signal but

unwanted.

• Classification of Signals:

Depending on the independent variables and the value of the

function defining the signal.

1. Continuous-Time (CT) and Discrete-Time(DT) Signals

2. Continuous-valued and Discrete-valued Signals

3. Multichannel and Multidimensional Signals

4. Deterministic and Random Signals

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Continuous-Time (CT) and Discrete-Time (DT) Signals:

• Continuous-Time (CT) Signal:

➢ A CT Signal is a signal that is defined at each and every instant of time.

It can be represented as x(t), where t is the independent variable.

➢ This type of signal shows continuity both in amplitude and time. These

will have values at each instant of time. Sine and cosine functions are

the best example of Continuous time signal.

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Continuous-Time (CT) and Discrete-Time (DT) Signals:

• Discrete-Time (DT) Signals:

➢ A DT signal is a signal that is defined at discrete instant of time. It can

be represented as x(nT), where n is an integer and T is the time interval

between two consecutive signal values (Sampling period).

➢ This type of signal shows continuity in amplitude but discrete in time.

➢ Relationship between time variables t and n of CT and DT signals.

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Representation of Discrete-Time (DT) Signals:

• Graphical Representation

• Functional Representation

• Tabular Representation n … -3 -2 -1 0 1 2 3 …

X[n] … 0 0 0 1 1 1 1 …

• Sequence Representation . . . 0 0 0 ณ1↑

1 1 1 . . .

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• Continuous Valued and Discrete Valued Signals:

➢ Values of CT or DT signals can be continuous or discrete.

➢ If the signal takes on all possible values on a finite or an

infinite range, it is said to be a Continuous valued signal.

➢ If the signal takes a set of discrete values, it is called Discrete

valued signal.

➢ Continuous time and continuous valued : Analog signal.

➢ Continuous time and discrete valued: Quantized signal.

➢ Discrete time and continuous valued: Sampled signal.

➢ Discrete time and discrete values: Digital signal.

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Multichannel and Multidimensional Signals:

• Multichannel Signal:

➢ Signal is generated from multiple sources.

➢ For example: Electrocardiography (ECG) 3 lead and 12 lead signal.

𝑥 𝑡 =

𝑥1(𝑡)𝑥2(𝑡)𝑥3(𝑡)

• Multidimensional Signal:

➢ If the signal is function of one independent variable is called one

dimension signal otherwise the signal is called M-dimensional signal

➢ For example: Video signal, I(x,y,t) is a 3-Dimensional signal because I is

the function of three independent variables (x,y,t).

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Deterministic and Random Signals:

• Deterministic Signal:

➢ A signal is said to be deterministic if there is no uncertainty with respect

to its value at any instant of time. Or, signals which can be defined

exactly by a mathematical formula are known as deterministic signals.

➢ This signal is predicted at any time.

• Random Signal:

➢ A signal is said to be Random if there is

uncertainty with respect to its value at some

instant of time

➢ Random signals cannot be described by a

mathematical equation.

➢ Random signals are modelled in probabilistic

terms.

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Standard Discrete-Time Signals:

• Unit Step Sequence:

• Unit Sample Sequence (Impulse Sequence):

➢ The unit step sequence can be written in terms of delayed impulses as

𝑢 𝑛 = 𝛿 𝑛 + 𝛿 𝑛 − 1 + 𝛿 𝑛 − 2 +⋯ = σ𝑘=0∞ 𝛿[𝑛 − 𝑘]

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Standard Discrete-Time Signals:

• Unit Ramp Sequence:

• Exponential Sequence:

➢ Exponential sequence are important in representing and analyzing liner time

invariant systems.

➢ An exponential signal can either have exponentially rising or falling

amplitude depending upon its exponent value.

➢ The general form of an exponential sequence is given by 𝑥 𝑛 = 𝛼𝑛 ∀ 𝑛 .

➢ If α is real then the sequence is real.

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• Exponential Sequence:

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• Exponential Sequence:

➢ When “α” is complex, a more general case to consider is the

complex exponential sequence:

𝑥 𝑛 = 𝐴𝛼𝑛 where 𝛼 = 𝛼 𝑒𝑗𝜔𝑜 and A = 𝐴 𝑒𝑗φ

𝑥 𝑛 = 𝐴𝛼𝑛 = 𝐴 𝑒𝑗φ |𝛼|𝑛𝑒𝑗𝜔𝑜𝑛

= 𝐴 |𝛼|𝑛𝑒𝑗(𝜔𝑜𝑛+φ)

= 𝐴 |𝛼|𝑛cos (𝜔𝑜𝑛 + φ) + 𝑗 𝐴 |𝛼|𝑛sin (𝜔𝑜𝑛 + φ)=Re 𝑥(𝑛) + 𝑗𝐼𝑚 𝑥(𝑛)

Polar form

➢ If 𝛼 < 1, the real and imaginary part of the sequence

magnitude oscillate with exponentially decaying envelopes.

➢ If 𝛼 > 1, the real and imaginary part of the sequence

magnitude oscillate with exponentially growing envelopes.

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• Exponential Sequence:

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• Exponential Sequence:

➢ When 𝛼 =1, x(n) is referred to as the discrete-time complex

sinusoidal sequence and has the form:

𝑥(𝑛) = 𝐴 cos (𝜔𝑜𝑛 + φ) + 𝑗 𝐴 sin (𝜔𝑜𝑛 + φ)

➢ For complex sinusoidal sequence, the real and imaginary part of

the sequence magnitude oscillate with constant envelopes.

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Operation on Discrete-Time Signals:

• Signal processing is a group of basic operations applied to an

input signal resulting in another signal as the output.

• The basic set of operations are:

➢ Time Shifting

➢ Time Scaling

➢ Time Reversal

➢ Signal Multiplier

➢ Signal Addition

OperationDT input

SignalX(n)

DT Output Signaly(n)

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Operation on Discrete-Time Signals:• Time Shifting: The name suggests, the shifting of a signal in time.

This is done by adding or subtracting an integer quantity of the shift

to the time variable in the function.

• Subtracting a fixed positive quantity from the time variable will shift

the signal to the right (delay) by the subtracted quantity.

• Adding a fixed positive amount to the time variable will shift the

signal to the left (advance) by the added quantity.

Delay kDT input

SignalX(n)

DT Output Signal

y(n)=x(n-k)

Advance kDT input

SignalX(n)

DT Output Signal

y(n)=x(n+k)

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Operation on Discrete-Time Signals:• Time Shifting (Delay): Right Shifting

Delay kDT input

SignalX(n)

DT Output Signal

y(n)=x(n-k)

𝑥[n]= 0 0.25 0.75 ณ1↑

0.75 0.25 0 𝑥[n-3]= ณ0↑

0.25 0.75 1 0.75 0.25 0

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Operation on Discrete-Time Signals:• Time Shifting (Advance): Left Shifting

Advance kDT input

SignalX(n)

DT Output Signal

y(n)=x(n+k)

𝑥[n]= ณ1↑

2 3 4 𝑥[n+1]= 1 ณ2↑

3 4

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Operation on Discrete-Time Signals:• Time Scaling: Time scaling compresses or dilates a signal by

multiplying the time variable by some quantity.

• If the quantity is greater than one, the signal becomes narrower and

the operation is called decimation.

• If the quantity is less than one, the signal becomes wider and the

operation is called expansion or interpolation, depending on how the

gaps between values are filled.

Time Scaling

k=2

DT input SignalX(n)

DT Output Signal

y(n)=x(2n)

Time Scaling

k=1/2

DT input SignalX(n)

DT Output Signal

y(n)=x(n/2)

Compress the signal x(n)

Expand the signal x(n)

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Operation on Discrete-Time Signals:• Time Scaling (Compress): Signal becomes narrower

Time Scaling

k=2

DT input SignalX(n)

DT Output Signal

y(n)=x(2n)

𝑥[n]= 0 0.25 0.75 ณ1↑

0.75 0.25 0 𝑥[2n]= 0 0 0.25 ณ1↑

0.25 0 0

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Operation on Discrete-Time Signals:• Time Scaling (Expand): Signal becomes narrower

𝑥[n]= 0 0.25 0.75 ณ1↑

0.75 0.25 0 𝑥[2n]= 0 0 0.25 ณ1↑

0.25 0 0

Time Scaling

k=1/2

DT input SignalX(n)

DT Output Signal

y(n)=x(n/2)

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Operation on Discrete-Time Signals:• Time Reversal: This operation is the reversal of the time axis, or

flipping the signal over the y-axis.

• Folding the sequence x[n] about n=0.

• Mathematically, it is expressed as x[-n]

Time ReversalDT input

SignalX(n)

DT Output Signal

y(n)=x(-n)

𝑥[n]= −3 − 2 − 1 ณ0↑

1 2 3 𝑥[2n]= 3 2 1 ณ0↑

− 1 − 2 − 3

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Operation on Discrete-Time Signals:

• Time –Scaling and Time- Shifting operations are not

commutative.

• Time-Reversal and Time-Shifting operations are not

commutative.

• Time –Scaling and Time-Reversal operations are commutative.

• All above operations are based on transformations of the

independent variable i.e., discrete time n.

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Sequence of Operations:

Step 1: First delay or advance the signal i.e., first operation is the

Time-Shifting.

Step 2: Perform Time-Scaling and/or Time-Reversal on the shifted

signal

𝑄𝑢𝑒𝑠𝑡𝑖𝑜𝑛: 𝑆𝑘𝑒𝑡𝑐ℎ 𝑥 −𝑛 + 2 𝑎𝑛𝑑 𝑥 −𝑛 − 2 𝑤ℎ𝑒𝑛 𝑥 𝑛 = ณ2↑

1 3 5 8

𝑆ℎ𝑖𝑓𝑡 𝑏𝑦 𝑘 𝑓𝑜𝑙𝑑 𝑥(𝑛) = 𝑆ℎ𝑖𝑓𝑡 𝑏𝑦 𝑘 𝑥(−𝑛) = 𝑥(−𝑛 + 𝑘)

𝐹𝑜𝑙𝑑 𝑆ℎ𝑖𝑓𝑡 𝑏𝑦 𝑘 𝑥(𝑛) = 𝐹𝑜𝑙𝑑 𝑥(𝑛 − 𝑘) = 𝑥(−𝑛 − 𝑘)

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Operation on Discrete-Time Signals:

• Scalar Multiplication (Amplitude Scaling): The signal x(n) is

multiplied by a scalar factor ‘a’.

DT input SignalX(n)

DT Output Signal

y(n)=a. x(n)

a

If 𝑥 𝑛 = ณ2↑

1 3 5 8 and a = 2 then 𝑦 𝑛 = ณ4↑

2 6 10 16

• For Example:

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Operation on Discrete-Time Signals:

• Signal Multiplier: Multiplication of two signals to form

another sequence.

DT input SignalX(n)

DT Output Signal

y(n)=a. x(n)

a

If 𝑥 𝑛 = ณ2↑

1 3 5 8 and a = 2 then 𝑦 𝑛 = ณ4↑

2 6 10 16

• For Example:

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Sampling Process

• To be able to process a continuous valued continuous-time i.e

analog signal by a digital processor, we must first sample it to

generate a discrete-time signal the quantize it to get a quantized

discrete-time signal.

• A sampling system comprises three main components:

✓ Sampler

✓ Quantizer

✓ Encoder

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Sampling Process• Sampling is defined as, “The process of measuring the instantaneous values

of continuous-time signal in a discrete form.”

• Sample is a piece of data taken from the whole data which is continuous in

the time domain.

• When a source generates an analog signal and if that has to be digitized,

having 1s and 0s i.e., High or Low, the signal has to be discretized in time.

This discretization of analog signal is called as Sampling.

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Sampling Process• To discretize the signals, the gap between the samples should be fixed. That gap

can be termed as a sampling period Ts.

• 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 =1

𝑇𝑠= 𝑓𝑠

• The sampling rate fs denotes the number of samples taken per second, or for a

finite set of values.

• For an analog signal to be reconstructed from the digitized signal, the sampling

rate should be highly considered. The rate of sampling should be such that the

data in the message signal should neither be lost nor it should get over-lapped.

Hence, a rate was fixed for this, called as Nyquist rate.

• The sampling theorem, which is also called as Nyquist theorem, delivers the

theory of sufficient sample rate in terms of bandwidth for the class of functions

that are bandlimited.

• 𝑓𝑠 = 2𝐵

• The sampling theorem states that, “a signal can be exactly reproduced if it is

sampled at the rate fs which is greater than twice the maximum frequency B.”