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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
Digital Signal Processing
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:
Digital Signal Processing
A/D ConverterDigital SignalProcessing
D/A Converter
Analog input signal
Analog outputsignal
Digital input signal
Digital output signal
1.1. Signals, Systems and Signal Processing
Digital Signal Processing
1.1. Signals, Systems and Signal Processing
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.
Digital Signal Processing
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:
Digital Signal Processing
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. Classification of Signals
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.
Digital Signal Processing
x(n)
n0 1 2 3 4 5 6 7-1 8
Digital Signal with 4 different amplitude values
1.2. Classification of Signals
1.2.4. Deterministic versus Random Signals:
Digital Signal Processing
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
Digital Signal Processing
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:
1.4. A/D & D/A Conversion
Digital Signal Processing
1.4.1. Analog to Digital Converter (A/D):
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:
1.4. A/D & D/A Conversion
Digital Signal Processing
1.4.1. Analog to Digital Converter (A/D):
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
Digital Signal Processing
1.4.1. Analog to Digital Converter (A/D):
Relation among frequency variable:
1
1.4.1.1. Sampling:
1.4. A/D & D/A Conversion
Digital Signal Processing
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:
1.4. A/D & D/A Conversion
Digital Signal Processing
1.4.1. Analog to Digital Converter (A/D):
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. A/D & D/A Conversion
Digital Signal Processing
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
1.4. A/D & D/A Conversion
Digital Signal Processing
1.4.1. Analog to Digital Converter (A/D):
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:
1.4. A/D & D/A Conversion
Digital Signal Processing
1.4.1. Analog to Digital Converter (A/D):
• 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:
1.4. A/D & D/A Conversion
Digital Signal Processing
1.4.1. Analog to Digital Converter (A/D):
Reconstruction Formula:
The signal:
The samples:
Formula:
Interpolation Function:
1.4. A/D & D/A Conversion
Digital Signal Processing
1.4.1. Analog to Digital Converter (A/D):
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.
1.4. A/D & D/A Conversion
Digital Signal Processing
1.4.1. Analog to Digital Converter (A/D):
1.4.1.2. Quantization:
Fs = 1 Hz
1.4. A/D & D/A Conversion
Digital Signal Processing
1.4.1. Analog to Digital Converter (A/D):
1.4.1.2. Quantization:
Numerical illustration of quantization with one significant digit using truncation or rounding
1.4. A/D & D/A Conversion
Digital Signal Processing
1.4.1. Analog to Digital Converter (A/D):
1.4.1.3. Coding:
IntroductionIntroduction1.4. A/D & D/A Conversion
Digital Signal Processing
1.4.2. Digital to Analog Converter (A/D):
1
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