Digital Signal Processing Analysis Of Discrete Time Linear .

Digital Signal ProcessingModule 1Analysis of Discrete time Linear Time - Invariant SystemsObjective:1. To understand the representation of Discrete time signals2. To analyze the causality and stability concepts of Linear Shift Invariant (LSI) systemsIntroduction:Digital signals are discrete in both time (the independent variable) and amplitude (thedependent variable). Signals that are discrete in time but continuous in amplitude are referredto as discrete-time signals.A discrete-time system is one that processes a discrete-time input sequence to producea discrete-time output sequence. There are many different kinds of such systems. One of theimportant kinds is Linear Shift Invariant (LSI) systems.Description:Discrete time Signals and SequencesDiscrete-time signals are data sequences. A sequence of data is denoted {x[n]} orsimply x[n] when the meaning is clear. The elements of the sequence are called samples. Theindex n associated with each sample is an integer. If appropriate, the range of n will bespecified. Quite often, we are interested in identifying the sample where n 0. This is doneby putting an arrow under that sample. For instance,𝑥𝑛 ,0.35,1,1.5, 0.6, 2, The arrow is often omitted if it is clear from the context which sample is x[0]. Sample valuescan either be real or complex. The terms “discrete-time signals” and “sequences” are usedinterchangeably.The time interval between samples is can be assumed to be normalized to 1 unit of time. Sothe corresponding normalized sampling frequency is 1Hz. If the actual sampling interval is T1seconds, then the sampling frequency is given by 𝑓𝑠 𝑇Some Elementary Sequencesi)Unit Impulse SequenceThe unit impulse sequence is defined byThis is depicted graphically in Figure 1.1. Note that while the continuous-time unit impulsefunction is a mathematical object that cannot be physically realized, the unit impulsesequence can easily be generated.

Figure 1.1: The Unit Impulse Sequenceii)Unit Step SequenceThe unit step sequence is one that has amplitude of zero for negative indices and amplitude ofone for non-negative indices.0 𝑛 0𝑢 𝑛 1 𝑛 0It is shown in Figure 1.2.Figure 1.2: The Unit Step Sequenceiii)Exponential sequencesThe general form isIf A and α are real numbers then the sequence is real. If 0 α 1 and A ispositive, then the sequence values are positive and decrease with increasing n:

iv)Sinusoidal SequencesA sinusoidal sequence has the formThis function can also be decomposed into its in-phase xi[q] and quadrature xq[n]components.This is a common practice in communications signal processing. It is shown in Figure 1.3.Figure 1.3: The Sinusoidal Sequencev)Complex Exponential SequencesComplex exponential sequences are essentially complex sinusoids.vi)Random SequencesThe sample values of a random sequence are randomly drawn from a certain probabilitydistribution. They are also called stochastic sequences. The two most commondistributions are the Gaussian (normal) distribution and the uniform distribution. Thezero-mean Gaussian distribution is often used to model noise. Figure 1.4 and Figure 1.5show examples of uniformly distributed and Gaussian distributed random sequencesrespectively.

Figure 1.4: Uniformly distributed random sequence with amplitudes between -0.5 and 0.5.Figure 1.5: Gaussian distributed random sequence with zero mean and unit variance.Types of SequencesThe discrete-time signals that we encounter can be classified in a number of ways. Somebasic classifications that are of interest to us are described below.i)ii)iii)Real vs. Complex SignalsA sequence is considered complex at least one sample is complex-valued.Finite vs. Infinite Length SignalsFinite length sequences are defined only for a range of indices, say N1 toN2. Thelength of this finite length sequence is given by N2- N1 1 .Causal SignalsA sequence x[n] is a causal sequence if x[n] 0 for n 0.

iv)Symmetric SignalsFirst consider a real-valued sequence {x[n]}.Even symmetry implies that x[n] x[-n] andOdd symmetry implies that x[n] -x[n] for all n.Any real-valued sequence can be decomposed into odd and even parts so thatwhere the even part is given byand the odd part is given byA complex-valued sequence is conjugate symmetric if x[n] x*[-n]. The sequence hasconjugate anti-symmetry if x[n] -x*[-n]. Analogous to real-valued sequences, any complexvalued sequence can be decomposed into its conjugate symmetric and conjugate antisymmetric parts:v)Periodic SignalsA discrete-time sequence is periodic with a period of N samples iffor all integer values of k. Note that N has to be a positive integer.If a sequence is not periodic, it is aperiodic or non-periodic.Consider a discrete-time sequence x [n] based on a sinusoid with angular frequency ωo:If this sequence is periodic with a period of N samples, then the following must be true:However, the left hand side can be expressed asand the cosine function is periodic with a period of 2π and therefore the right hand sideof the above equation is given byfor integer values of r.Comparing the above two equations, we have

where ωo 2πfo. Since both r and N are integers, a discrete-time sinusoidal sequence isperiodic if its frequency is a rational number. Otherwise, it is non-periodic.vi)Energy and Power SignalsThe energy of a finite length sequence x[n] is defined aswhile that for an infinite sequence isNote that the energy of an infinite length sequence may not be infinite. A signal with finiteenergy is usually referred to as an energy signal.The average power of a periodic sequence with a period of N samples is defined asand for non-periodic sequences, it is defined in terms of the following limit if it exists:A signal with finite average power is called a power signal.vii)Bounded SignalsA sequence is bounded if every sample of the sequence has a magnitude which is less than orequal to a finite positive value. That is,viii) Summable SignalsA sequence is absolutely summable if the sum of the absolute value of all its samples is finite.

A sequence is square summable if the sum of the magnitude squared of all its samples isfinite.Classification of SystemsDiscrete-time systems, like continuous-time systems, can be classified in a variety ofways.i)Linearity A linear system is one which obeys the superposition principle. For acertain system, let the outputs corresponding to inputs x1[n] and x2[n] are y1[n]and y2[n] respectively. Now if the input is given bywhere A and B are arbitrary constants, then the system is linear if its corresponding output isSuperposition is a very nice property which makes analysis much simpler. Linearization is avery useful approach to analyzing nonlinear systems.ii)Shift InvarianceA shift (or time) invariant system is one that does not change with time. Let a systemresponse to an input x[n] be y[n]. If the input is now shifted by n0 (an integer) samples,then the system is shift invariant if its response to x1[n] isWe can use the terms linear time-invariant (LTI) and linear shift-invariant interchangeably.iii)CausalityThe response of a causal system at any time depends only on the input at the current and pastinstants, not on any future samples. In other words, the output sample y[no] for any no onlydepends on x[n] for n n0.iv)StabilityThere are two common criteria for system stability. They are exponential stability andbounded-input bounded-output (BIBO) stability. It requires the response of the system todecay exponentially fast for a finite duration input. The second one merely requires that theoutput be a bounded sequence if the input is a bounded sequence.Linear Shift-Invariant SystemsA discrete-time LTI system, now called as LSI system, like its continuous-timecounterpart, is completely characterized by its impulse response. In other words, the impulseresponse tells us everything we need to know about an LSI system as far as signal processing

is concerned. The impulse response is simply the observed system output when the input is animpulse sequence. The output of the LSI system is using Convolution sum.Condition for StabilitySince the impulse response completely characterizes an LSI system, we draw conclusionsregarding the stability of a system based on its impulse response.Theorem 1.1.An LSI system is BIBO stable if its impulse response is absolutely summable.Proof:Let the input be bounded, i.e. x[n] B 1 for some finite value B. The magnitude of theoutput is given bySo the magnitude of y[n] is bounded ifresponse must be absolutely summable.is finite. In other words, the impulseCondition for CausalityTheorem 1.2.An LSI system is causal if and only if its impulse response is a causal sequence.Proof.Consider an LSI system with impulse response h[k]. Two different inputs x1[n] and x2[n] arethe same up to a certain point in time, that is x1[n] x2[n] for n n0 for some no. The outputsy1[n] and y2[n] at n no are given by

andSince x1[n] x2[n] for n no, if the system is causal, then the outputs y1[n] and y2[n] must bethe same for n no. More specifically, y1[no] y2[no]. Now,becausefor non-negative values of k. Since x1[n] may not beequal to x2[n] for n no, we must havewhich means that h[k] 0 for k 0.Illustrative Examples:Problem 1: Check whether the system defined by y(n) x2(n) is shift-invariant.Solution:If y(n) x2(n) is the response of the system to x(n), the response of the system tox'(n) x(n - no) isBecause y'(n) y(n - no), the system is shift-invariant.Problem 2: Check whether the systems described by the equationsi)y(n) x(n) x(n - 1 )ii)y(n) x(n) x(n 1)Solution:i)ii)y (n) x(n) x(n - 1 ) is causal because the value of the output at any time n nodepends only on the input x(n) at time no and at time no - 1.y(n) x(n) x(n 1) is noncausal because the output at time n no depends on thevalue of the input at time no 1.

Problem 3: An LSI system with unit sample response h(n) anu(n) will be stable whenever a 1. Justify.Solution:We have learnt that the magnitude of y[n] is bounded ifimpulse response must be absolutely summable.is finite i.e., theHence, the impulse response is absolutely summable only when a 1Problem 4: Prove that the system described by the equation y(n) n x(n) is not BIBO stable.Solution:Assume that the input signal is a unit step signal x(n) u(n) which is bounded.0 𝑛 0𝑢 𝑛 1 𝑛 0But the response to a unit step is y(n) n u(n), which is unbounded as n increasesHence the system is not BIBO stable.Problem 5: Find the response of the system with impulse response h(n) u(n) for the inputx(n) anu(n).Solution:With the direct evaluation of the convolution sum we find the response of the systemBecause u(k) is equal to zero for k 0 and u(n - k) is equal to zero for k n, when n 0,there are no nonzero terms in the sum and y(n) 0. On the other hand, if n 0,SinceTherefore,Summary:Discrete-time signals are essentially a sequence of numbers. Some fundamentalsequences, such as the unit impulse, unit step and sinusoidal (both real and complex)sequences are examined because more complex sequences can be expressed as a combinationof some or all of these fundamental ones.The most important discrete-time systems are the linear shift invariant (LSI) systems.For these systems, the superposition principle applies which leads to the linear convolutionsum. This convolution sum is the way by which we derive the output of the system given an