Part I-Introduction to Digital Signal Processing PDF

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Introduction to DSP

Int Intro ro rodu du ductio ctio ction n tto oD DSP SP In this section we will focus on:     

Signal properties and classifications Energy and power signals LTI systems Correlation Convolution

What is a sig sign nal?    

a function/physical quantity that conveys information about a phenomenon physical representation of information carried from its source to the receiver can be defined as electrical quantities varying to the time A signal typically can vary in time and/or in space. For example an image varies in space whilst a video varies both in time and space A signal constitutes of a measurable magnitude examples include current, voltage, force, temperature, pressure, finances, vibrations, etc.

What is n noi oi oise? se?  

Any disturbance/disturbing phenomenon that interferes with the perception or the interpretation of a signal Difference between a signal and noise depends solely on the interest of the user

What is ssign ign ignal al p pro ro roce ce cessin ssin ssing? g?   

Digital Signal Processing (DSP) is concerned with the representation, transformation and manipulation of signals on a computer. The purpose of signal processing is to make it easier to extract useful information and make correct decisions With this we have signal enhancement, signal restoration, signal reconstruction, signal synthesis, signal estimation and prediction

What is d digit igit igitali ali alisa sa sati ti tion on on??  

Changing of a signal from the continuous times and amplitudes to the discrete times and amplitudes domain Mainly broken into 2 categories: sampling and quantization

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Why d dig ig igital ital sig signa na nals? ls?      

Greater noise immunity Digital circuit components are cheaper and easily produced Uses less bandwidth Can be encrypted, only desired receiver can receive the signal thus more security Simple to analyse Complicated processes are achievable when using digital signals than analogue

Sig Signa na nall Cla Class ss ssific ific ificatio atio ation ns Continuous time vs. discrete time  

Continuous time signals are signals defined continuously over time Discrete time signals are signals defined at a discrete set of time signals

Real signals vs. complex signals  

Real signals can either be continuous or discrete signals which takes a set of real numbers Complex signals take values belonging to a set of complex numbers

Deterministic vs. random signals  

deterministic signals are completely specified at any given time instant random signals take values of different time instants, for example outcome of a coin toss

Even vs. odd signals  

a signal is said to be even when it satisfies the condition a signal is said to be odd when it satisfies the condition

Real and complex signals  

real signals can either be continuous or discrete signals which takes a set of real numbers; complex signals take values belonging to a set of complex numbers;

Periodic vs. aperiodic signals  

a continuous signal is periodic if there exists a time period T such that for all values of t, otherwise it is aperiodic a discrete signal is periodic if there exists a time period N such that for all values of n

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Imp Impor or ortan tan tantt sig sign nals to b be e us used ed du durin rin ringg cco ours urse eo off se seme me mest st ster er Unit impulse function/Dirac delta function

Unit step function

Sign function

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Rectangular function

En Energ erg ergyy sig sign nals and po power wer sig sign nals  

An energy signal has a finite energy and its power is zero(E=finite and P=0) A power signal has infinite energy and finite power (E=∞ and P=finite)

Cla Classif ssif ssifica ica icatio tio tion no off ssys ys ystem tem temss 

a system representation is a mathematical model for a physical process, or a device that describes the output signal for any given input signal

Memory and memory less systems  

in memory less systems, the output depends only on current inputs at an instant in memory systems, output depends not only on the current input but also past values

Causal and non-causal systems  

For a causal system, the output depends only on the inputs where . The output depends only on the past values of the input For an anti-causal system, the output not only depends on the past inputs but also future values of the input signal

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Stable system 

A LTI system with impulse response h(n) is bounded if and only if S is defined as follows

Lin Linear ear Ti Timememe-Inv Inv Invar ar aria ia iant nt ((LT LT LTI) I) SSyste yste ystem ms 

A linear system is a system which satisfies both additivity and homogeneity, otherwise it is a non-linear system

Linearity 

A system T is said to be linear if for every and (either real or complex) and every input signal and it holds that

Time-Invariance 

A system T is said to be time-variant if and only if

Causality 

A system T is said to be causal if, for any n,

.

Stability  

BIBO-bounded input bounded output, is the criterion for stability A LTI system with impulse response h(n) is bounded if and only if S is defined as follows

Sys System tem d des es escrib crib cribed ed b byy a d dif if iffer fer ferenc enc ence e eequ qu quaatio tion n



If the coefficients , the system is said to be recursive thus IIR

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

If the coefficients , then the system is dependant only on and it is said to be non recursive thus FIR

Cor Corre re relatio latio lation n an and d au auto to to-co -co -correla rrela rrelatio tio tion n 

Correlation computes a measure of similarity of 2 input signals as they are shifted by one another Suppose x(n) and y(n), two signals of finite energy and inter-correlation



Inter-correlation between x(n) and y(n) has its maximum for a delay k if



Auto-correlation defines correlating a signal with itself.



Properties of autocorrelation 

For

we find the energy of the signal, which is the maximum correlation thus

  

is maximum for If x(n) is real, the autocorrelation is real and paired Autocorrelation of a signal of size N will have a sixe of

Co Convo nvo nvolu lu lutio tio tion n  

A system is characterised by its impulse response We can deduce the effect of any entry in the form of a convolution

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