Department of Computer Engineering
SKNCOE Vadgaon Pune-41
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Name of Topics
What is DSP? The Breadth and Depth of DSP.
What is Signals ? Classification of signals
How signals is created?
ADC and DAC, sampling,
Statistics, probability and noise
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Discrete time system
Properties of DT system
Mathematical model for representation of DT system
Linear system
Use of transducers in DSP
Name of books
Referred
Steven W. Smith
Digital Signal Proc.
John Proakis
Digital Signal Proc.
John Proakis
Digital Signal Proc.
Steven W. Smith
Digital Signal Proc.
John Proakis
Digital Signal Proc.
John Proakis
Digital Signal Proc.
Steven W. Smith
Digital Signal Proc.
Steven W. Smith
Digital Signal Proc.
No of
Lecture
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D-(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 0
S-(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.
Analog o/P signals
Analog I/P signals
A to D convertor
Digital I/P signals
Digital signal processor
Digital o/P signals
D to A
Convertor
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 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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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/retrieval
Commercial
 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
 Filtering
Military
 Radar
 Sonar
 Ordnance guidance
 Secure communication
Industrial
 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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 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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Signals are classified according to their characteristics
1. Continuous time and discrete time signals
2. Deterministic and random
3. Periodic and non periodic signals
4. Even and odd signals
5. Energy and power signals
6. Causal and non causal signals
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 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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 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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 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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 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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 A continuous time signal is said to be symmetric (even) if it satisfies the condition
 A continuous time signal is said to be anti- symmetric (odd) if it satisfies the condition
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 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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 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 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
 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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 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 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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 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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 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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Analog signal x(t)
Sampler
Discrete time signals x(n)
Quantizer
Quantized signal xq(n)
Encoder
Digital signal
X[n]
Analog signal x(t)
Fs=1/T
X(n)=x(nT)
Discrete time signal
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 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.
Digital signal Interpolator
Filter analog signal
Basic block diagram of DAC
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 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.
X[n] input signal
Discrete time system
Y[n] output signal
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 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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 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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 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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 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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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 equation
Condition
 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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 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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 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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 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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Input device
Microphone
Controller/System
Amplifier
Output device loudspeaker
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.
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