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Prof. Virendra V. DakhodeDepartment of Computer Engineering

SKNCOE Vadgaon Pune-41

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Sr. No

Name of Topics Name of booksReferred

No of Lecture

1 What is DSP? The Breadth and Depth of DSP.

Steven W. SmithDigital Signal Proc.

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2 What is Signals ? Classification of signals

John ProakisDigital Signal Proc.

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3 How signals is created?ADC and DAC, sampling,

John ProakisDigital Signal Proc.

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4 Statistics, probability and noise Steven W. SmithDigital Signal Proc.

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5 Discrete time systemProperties of DT system

John ProakisDigital Signal Proc.

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6 Mathematical model for representation of DT system

John ProakisDigital Signal Proc.

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7 Linear system Steven W. SmithDigital Signal Proc.

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8 Use of transducers in DSP Steven W. SmithDigital Signal Proc.

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What is DSPD-(Digital):-Digital generates store and process data in term of two

state –ve and +ve.+ve is express as represent by 1-ve is express as represent by 0S-(Signal):- A signal is defined as any physical quantity that varies

with time, space or any other independent variable or variable .P-(Processing):-To perform operation on data according to

programmed instruction.

A to D convertor

Digital signal processor

D to A Convertor

Analog I/P signals

Digital I/P signals

Digital o/P signals

Analog o/P signals

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Allied areas of DSP Telecommunication (telephone conversion, telephone signals) 1. Multiplexing 2. Compression 3. Echo control Audio Processing 1. Music 2. Speech recognition 3.Speech generation Echo location 1. Radar (Radio detection & ranging) 2.Sonar (Sound navigation & ranging) 3. Reflection seismology

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Image processing 1. Medical 2. Space 3. Commercial imaging product.

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The Breadth and Depth of DSP

Space Space photograph enhancement Data compression Intelligent sensory analysis by remote space probes Medical Diagnostic imaging (CT, MRI, ultrasound, and others) Electrocardiogram analysis Medical image storage/retrievalCommercial Image and sound compression for multimedia presentation Movie special effects Video conference calling

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Telephone Voice and data compression Echo reduction Signal multiplexing FilteringMilitary Radar Sonar Ordnance guidance Secure communicationIndustrial Oil and mineral prospecting Process monitoring & control Non destructive testing CAD and design tools

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Scientific Earthquake recording & analysis Data acquisition Spectral analysis Simulation and modelling

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What is signal

A signal is defined as a physical quantity that varies with time, space or any other independent variable

The signal may depend on one or more independent variable. If a signal depends on only one variable then it is known as

one dimensional signal. Ex. AC power signal, speech signal ,ECG signal etc. If a signal depends on two independent variable then the

signal is known as two dimensional signals. Ex. X-ray , sonograms. Multi dimensional signal depends on many variables.

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Classification of signals

Signals are classified according to their characteristics1. Continuous time and discrete time signals2. Deterministic and random 3. Periodic and non periodic signals4. Even and odd signals5. Energy and power signals6. Causal and non causal signals

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Continuous time signals

Continuous time signals are defined for all values of “t” and is represented by x(t) .

Continuous time signals is also called an analog signals.

Ex. AC power supply

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Discrete time signals

The discrete time signals are defined at discrete instance of time and represented by x(n).

Ex. The amount deposited every month in a savings account is discrete.

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Deterministic and random signals

A deterministic signal is a signal having certainty of values at any given instance of time. (In medical images like ECG)

A random signal is a signal having uncertainty before its actual occurrence.(noise, seismic signals)

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Periodic and non-periodic signals

A continuous time signal is said to be periodic if it satisfies the condition

x(t + T) = x(t) for all “t”

A discrete time signal is said to be periodic if it satisfies the condition

x(n) = x (n + N) for all “N”

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Symmetric (even) and anti- symmetric (odd)

A continuous time signal is said to be symmetric (even) if it satisfies the condition

x(-t) = x(t) for all “t” A continuous time signal is said to be anti- symmetric (odd) if it satisfies the condition x(-t) = - x(t) for all “t”

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Energy and Power Signals The total energy of a sequence of x[n] is defined by

An infinite length sequence with finite sample values may or may not have finite energy.

The average power of signal given by

Average power of an infinite length sequence may be finite or infinite.

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Causal and non-causal Signals

A continuous time signal x(f) is said to be causal if

X(f)=0 for t<0

Other wise it is non causal

Discrete time signal is said to be causal if X(n)=0 for n<0

Otherwise it is non casual

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Statistics, probability and noise

Statistics and probability are used in DSP to characterize signals and processes that generate them.

The primary use of DSP is to avoid interference, noise and other undesirable components in the acquired data.

All these are produced as unavoidable by product of some DSP operation.

Statistics and probability allows these disruptive features to be measured and classified and to remove that offending components.

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Mean and standard deviation

Mean

Mean indicated by μ=

In words sum the values in the signal Where “i” is the index run from 0 to N-1 and then divide the

sum by N. This identical to the equation

In electronics, mean is commonly called the DC(direct current)

value the AC (alternating current) refers to now the signal fluctuate around the mean values.

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Standard deviation

It is denoted by In equation form it is given by

The term occurs frequently in statistics and given the name variance.

Standard deviation is a measure of how the away the signal fluctuate from the mean.

Variance represents the power of this fluctuation. Mean describes what is being measured. Standard deviation represents noise and other interference.

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The histogram, PMF and PDF

The Histogram display the number of samples that are in the signal that have each of the possible values.

The sum of all values in the histogram is equal to the number of points in the signal

Where Hi is the signal N is the number of points in signal M is number of points in histogram The histogram can be used to efficiency calculate the mean

and std. Deviation of very large data sets. This is especially important for image which can millions of

samples.

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Histogram groups samples together that have the same values.

Calculation of mean from histogram

Calculation of standard deviation from histogram

Limitation of histogram Calculating mean and standard deviation is time consuming

operations of addition and multiplication. Histogram algorithm is uses only on few samples.

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Probability mass function(PMF) A histogram is always calculated using a finite numbers of

samples while PMF is used with an infinite number of samples.

PMF is use for discrete signals. PDF is use for continuous signals.

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ADC and DAC Analog to digital conversion (ADC) and digital to analog

conversion (DAC) are the processes that allow the digital computers to interact with everyday signals.

Digital information is different form its continuous counterpart in two important respect it is sampled and quantized.

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ADC (Analog to digital conversion) The analog signal get convert into digital signal by performing

following operation like Sampling, quantization and encoding.

Most of the analog signal in the form of continuous time signal but in digital signal processing the signal are sampled and quantized at discrete time instance and represented by 0

and 1.This can be done by analog to digital convertor. Sampling: This is the conversion of a continuous time signal

into discrete time signal.

Quantization: this is the conversion of a discrete time continuous valued signal into a discrete time discrete value signal(digital signal)

Encoding: In the coding process each discrete value represented by binary sequence.

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ADC

Sampler Quantizer Encoder

Analog signalx(t)

Discrete time signals x(n)

Quantized signal xq(n)

Digital signalX[n]

Fs=1/TAnalog signalx(t)

Discrete time signal

X(n)=x(nT)

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DAC

To convert a digital signal into an analog signal A to D convertor is used.

In D to A convertor interpolation of samples performed. In interpolation it connects successive samples with straight

line segment D to A converter involves a sub optimum interpolator

followed by post filter.

Interpolator FilterDigital signal

analog signal

Basic block diagram of DAC

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Discrete time system Discrete time signals defined at discrete instance of time and

represented as x(n). Discrete time system is a device or algorithm that operates on

a discrete time signals. DT system processes a given input x[n] to generate an output

response with more desirable properties. In most of application discrete time system is a single input

single output system. Various types of discrete time systems are available science

the digital computer such as systems used for digital control, robotices,data compression and image processing.

Discrete time systemX[n] input signal Y[n] output signal

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Basic properties of discrete time systems

Linearity Time invariant Causality Static and dynamic system.

Linear system The system is linear if and only if it satisfies superposition

principal that is

If it does not satisfy above condition then system is said to be non linear

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Time invariant A system is said to be time invariant if its input output

characteristics does not change with time

Suppose we have a system T in relax state which, when exited by an input signal x(n) produces an output signal y(n) i.e.

Suppose we delay the input signal by ‘k’ units i.e. X(n-k) then

If the time of system do not change with time the output of the system is same i.e. Y(n-k) then the system is said to be time invariant/shift invariant otherwise time variant system.

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There fore, A relaxed system (time invariant )Or shift invariant if and only if

Implies that,

For every input signal x(n) and every time shift k.

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Causal system

A system is said to be causal if the output of the system at any time n [i.e. y(n) depends only on present and past input i.e. X(n),x(n-1),x(n-2).......] and does not depends on future input that is [x(n+1),x(n+2).......] ,

That is system is causal if it satisfy

If the system does not satisfy this question then it is said to be non causal.

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Static system A discrete time system called static/memory less system if its

output at any instant “n” depends at most on the same time , but not on past and future samples of the input otherwise system is said to be dynamic.

And

Both are said to be causal.

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Mathematical model for representation of DT system

Linear constant coefficient difference equation Difference equation describe a relationships between the

input and output rather than an explicit expression for the system output as a function of its input.

A linear constant coefficient difference equation of order N looks like

All solution of y[n] can be expressed as a sum.

Equation 1 can be rewritten as

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We need to know the input for all ‘n’ as well as a set of ‘N’ auxiliary condition such as

In order to be solve equationCondition An input x[n]=0 for leads to output y[n]=0 for A causal input x[n]=0 for n<0 leads to a causal output y[n]=0 for n<0.

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Linear convolution Consider unit step input x(n)=u(n) and filter Filtering is the operation of convolving a signal with the filter

impulse response.

Y(n)=0 , n<0

Y(0)=x(0).h(0)=1 (all other terms are zero)

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Linear System A signal is any physical quantity that carries information. OR we can say signal is a description of how one parameter

varies with anther parameter. A system is any process that produces an output signal in

response to an input signal. Linearity : A system is called linear if it has two mathematical

properties homogeneity and additive. Or a system is said to be linear if it obeys superposition

theorem. Homogeneity: It means that a change in amplitude of input

signal results in change in amplitude of output signal. x[n]=y[n] k x[n]=k y[n] where k is any constt.

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Additive: A system is said to be additive if added signal pass through it without interfacing.

If x1[n] result in y1[n] If x2[n] result in y2[n] then x1[n]+x2[n]=y1[n]+y2[n]Examples of linear systems Wave propagation: Electromagnetic waves Electrical circuits: resister, capacitor, inductor. Electronic circuits: Amplifiers and filters. Unit system: Where output is equal to input signal.

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Use of transducers in signal processing

Transducers defined as the device which convert one form of energy into other form.

The word transducer is a collective term used for both sensors and actuator.

Sensors which can be used to sense a wide range of different energy forms such as movement electrical signals, radiant energy, thermal energy.

Actuator used to switch voltage or current. Example 1: Microphone converts sound waves into electrical

signals for the amplifiers to amplify. Example 2: Loudspeaker(output device) convert these

electrical signals back into sound.

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1. Thermocouple used to produce an analog signal.2. Light sensor used to produce digital signal.3. Carbon microphone and piezo electric crystal are used to

measure sound.4. Thermister/thermostat are used to measure temperature

and many more.

AmplifierInput deviceMicrophone

Controller/SystemOutput device loudspeaker