digital signal processing prepared by: bhawna bhardwaj assistant professor b.p.r.c.e gohana

Digital Signal ProcessingDigital Signal ProcessingPrepared by:Bhawna BhardwajAssistant professorB.P.R.C.E Gohana

Digital Signal ProcessingCourse at a glance

Discrete-TimeSignals &Systems

Fourier DomainRepresentation

Sampling &Reconstruction

SystemStructure

SystemAnalysis

System

Z-Transform DFT

Filter

Filter Structure Filter Design

Chapter 1

Digital Signal Processing

Signals, Systems and Signal Processing.

Classification of Signals.

Concept of Frequency in Continuous-

Time & Discrete-Time Signals.

Analog to Digital & Digital to Analog

Conversion.

Fourier transform

1.1. Signals, Systems and Signal Processing

Signal is defined as any physical quantity that varies with independent

variables. For Example, the functions

S1(t) = 5t or S2(t) = 20t2 one variable

S(x,y) = 3x+4xy+6x2 two variables x and y

Speech signal


N

iiii ttFtA

1

))()(2sin()(

Amplitude Frequency Phase

System, is defined as a physical device that performs an operation on a

signal.

Basic elements of a digital signal processing system:


A/D ConverterDigital SignalProcessing

D/A Converter

Analog input signal

Analog outputsignal

Digital input signal

Digital output signal




Advantages of DSP

Flexibility (software change)

Accuracy

Reliable Storage

Complex process realized by simple code

Cost, Cheaper than analog

ttx cos)(1 tetx )(2

Classification of Signals

Continuous-Time versus Discrete-Time Signals:

Continuous-Time or analog signal are defined for every value of time.


are examples of analog signals

x(t)

t0 Analog Signal• Continuous in time. • Amplitude may take on any value in the continuous range of (-∞, ∞).

Analog Processing• Differentiation, Integration, Filtering, Amplification.• Implemented via passive or active electronic circuitry.

1.2. Classification of Signals

1.2.2. Continuous-Time versus Discrete-Time Signals:


Discrete-Time signals are defined only at certain specific value of time.

• Continuous Amplitude.• Only defined for certain time instances.• Can be obtained from analog signals via sampling.

The function provide an example of a discrete-time signal.

x(n)

n0 1 2 3 4 5 6 7-1

Undefined

Defined


1.2.3. Continuous-Valued versus Discrete-Valued Signals:The values of a CT or DT Signal can be continuous or discrete.If a signal takes on all possible values of a finite or an infinite range, it is CONTINUOUS-VALUED Signal.If the signal takes on values from a finite set of possible values, it is DISCRETE-VALUED Signal. Also called Digital SignalDigital Signal because of the discrete values.


x(n)

n0 1 2 3 4 5 6 7-1 8

Digital Signal with 4 different amplitude values


1.2.4. Deterministic versus Random Signals:


Random SignalA signal in which cannot be approximated by a formula to a

reasonable degree of accuracy (i.e. noise).

Deterministic SignalAny signal whose past, present and future values are

precisely known without any uncertainty

Fourier TransformFourier Transform• A CT signal A CT signal xx((tt) and its frequency domain, Fourier ) and its frequency domain, Fourier

transform signal, transform signal, XX((jj), are related by), are related by

• For example:For example:

• Often you have tables for common Fourier Often you have tables for common Fourier transformstransforms

• The Fourier transform, The Fourier transform, XX((jj), represents the ), represents the frequency content frequency content of of xx((tt).).

dejXtx

dtetxjX

tj

tj

)()(

)()(

21

)()( jXtxF

jatue

Fat

1

)(

analysis

synthesis

Fourier Transform of a Time Fourier Transform of a Time Shifted SignalShifted Signal• We’ll show that a Fourier transform of a signal which has We’ll show that a Fourier transform of a signal which has

a a simple time shift simple time shift is:is:

• i.e. the original Fourier transform but i.e. the original Fourier transform but shifted in phaseshifted in phase by –by –tt00

• ProofProof• Consider the Fourier transform synthesis equation:Consider the Fourier transform synthesis equation:

• but this is the synthesis equation for the Fourier but this is the synthesis equation for the Fourier transform transform

• ee--jj00ttXX((jj))

0

0

12

( )10 2

12

( ) ( )

( ) ( )

( )

j t

j t t

j t j t

x t X j e d

x t t X j e d

e X j e d

)()}({ 00 jXettxF tj

Convolution in the FrequencyConvolution in the Frequency DomainDomain• We can easily solve ODEs in the frequency We can easily solve ODEs in the frequency

domain:domain:

• Therefore, to apply Therefore, to apply convolution in the convolution in the frequency domainfrequency domain, we just have to , we just have to multiply multiply the the two Fourier Transformstwo Fourier Transforms..

• To solve for the differential/convolution equation To solve for the differential/convolution equation using Fourier transforms:using Fourier transforms:

1.1. Calculate Calculate Fourier transformsFourier transforms of of xx((tt) and ) and hh((tt): ): XX((jj) by ) by HH((jj))

2.2. MultiplyMultiply HH((jj) by ) by XX((jj) to obtain ) to obtain YY((jj))

3.3. Calculate the Calculate the inverse Fourier transforminverse Fourier transform of of YY((jj))

)()()()(*)()( jXjHjYtxthtyF

Proof of Convolution Proof of Convolution PropertyProperty

• Taking Fourier transforms gives:Taking Fourier transforms gives:

• Interchanging the order of integration, we haveInterchanging the order of integration, we have

• By the time shift property, the bracketed term is By the time shift property, the bracketed term is ee--

jjHH((jj), so), so

dthxty )()()(

dtedthxjY tj )()()(

ddtethxjY tj)()()(

)()(

)()(

)()()(

jXjH

dexjH

djHexjY

j

j

1.4. A/D & D/A Conversion


Sampler Quantizer Coder

Analogsignal

Digitalsignal

Discrete-Time signal

Quantized signal

x(n)

xa(t)

xq (n)

0101101…..

A/D Converter

1.4.1. Analog to Digital Converter (A/D):

Conceptually, the A/D comprise 3 step process as in the following figure.

1.4.1.1. Sampling:




It is the conversion of a CT signal into DT signal obtained by taking “Samples” of the CT signal at DT instants.

Periodic or Uniform Sampling:This type of sampling is used most often in practice, describe by the relation:

where x(n) is the DT signal obtained by taking samples of the analog signal xa(t) every T seconds.

The rate at which the signal is sampled is Fs: Fs = 1/T

Fs is called the SAMPLING RATE or SAMPLING FREQUENCY (Hz)

1.4.1.1. Sampling:




Consider an analog sinusoidal signal of the form:

Sampling Frequency:

Normalized frequency:

Sampled Signal:

1.4.1.1. Sampling:

IntroductionIntroduction1.4. A/D & D/A Conversion



Relation among frequency variable:

1

1.4.1.1. Sampling:



4.1. Analog to Digital Converter (A/D):

We observe that the fundamental difference between CT and DT signals in their range of values of the frequency variables F and f or Ω and ω.

Means Sampling from infinite frequency range for F (or Ω) into a finite frequency range for f (or ω).

Since the highest frequency in a DT signal is ω = π or f = 1/2.

With sampling rate Fs the corresponding highest values of F and Ω are:

1.4.1.1. Sampling:




Examples:

I. Two analog sinusoidal signals:

Which are sampled at a rate Fs = 40 Hz.

Discrete-time signals:

This mean

However,

The frequency F2 = 50 Hz is an alias of the frequency F1 = 10 Hz at the sampling rate of 40 samples per second.

F2 is not the only alias of F1



1.4.1. Analog to Digital Converter (A/D):II. Two analog sinusoidal signals, F1 = 1 Hz & F2 = 5 Hz are sampled at a rate Fs = 4 Hz.

F2 is the alias of F1




Aliasing

• Aliasing occurs when input frequencies (again greater than half the sampling rate) are folded and superimposed onto other existing frequencies.

In order to prevent alias

where Fmax is the highest input frequency

Nyquist Rate:

Minimum sampling rate to prevent alias.

1.4.1.1. Sampling:




• Given Band Limited (Frequency Limited Given Band Limited (Frequency Limited Signal) with highest frequency FSignal) with highest frequency Fmaxmax::The signal can be exactly reconstructed The signal can be exactly reconstructed provided the following is satisfied:provided the following is satisfied:– Sampling Frequency:Sampling Frequency:– The samples are not quantized The samples are not quantized

(analog amplitudes)(analog amplitudes)

Sampling Theorem:




Reconstruction Formula:

The signal:

The samples:

Formula:

Interpolation Function:




1.4.1.2. Quantization:

The process of converting a DT continuous amplitude signal into

digital signal by expressing each sample value as a finite number of

digits is called QUANTIZATION.





Fs = 1 Hz





Numerical illustration of quantization with one significant digit using truncation or rounding




1.4.1.3. Coding:

IntroductionIntroduction1.4. A/D & D/A Conversion


1.4.2. Digital to Analog Converter (A/D):

1

Thank youThank you

digital signal processing prepared by: bhawna bhardwaj assistant professor b.p.r.c.e gohana

Documents

dt signal

continuousvalued signal

digital digital

discretevalued signals

random signals

discrete values

simple time shift

certain time instances