Discrete-time Signals and Systems

SIGNALS, SPECTRA, AND SIGNAL PROCESSING

DISCRETE-TIME SIGNALS AND SYSTEMS


Objectives:1. To determine the characterization of discrete-time

systems in general and the class of linear time-invariant (LTI) systems in particular

2. To define and determine important time-domain properties of LTI systems

3. To derive the convolution formula, which allows us to determine the output of the LTI to any given arbitrary input signal

4. To introduced difference equations as an alternative method for describing the input-output relationship of an LTI system

Discrete-Time Signals


Alternative representation of a discrete-time signal or sequence:

1. Functional representation

2. Tabular representation

3. Sequence representation


Some Elementary Discrete-Time Signals1. Unit sample sequence


2. Unit step signal


3. Unit ramp signal


4. Exponential signal

if

then



(real part)


(imaginary part)


(amplitude function)


(phase function)

Classification of Discrete-Time Signals1. Energy signals and power signals

normalized energy:

if E is finite, then x(n) is called an energy signalaverage power,


but

then

if P is finite (and nonzero), the signal is called a power signal.


2. Periodic and aperiodic signalsA signal x(n) is periodic with period N (N > 0) if and only if

If the equation is not satisfied for any integer N, x(n) is said to be an aperiodic signal.


3. Symmetric (even) and antisymmetric (odd) signalsA real-valued signal x(n) is called symmetric (even) if


A signal x(n) is called antisymmetric (odd) if


Any signal x(n) may be decomposed into a sum of its even part, xe(n), and its odd part, xo(n), as follows:

To find the even part of x(n) we form the sum

To find the odd part of x(n) we form the difference


Signal ManipulationsTransformation of the Independent VariableSequences are often altered and manipulated by modifying the index n as follows:

where f(n) is some function of n.


1. ShiftingThis is the transformation defined by f(n) = n-no. If y(n) = x(n-no), x(n) is shifted to the right by nosamples if no is positive (this is referred to as a delay), and it is shifted to the left by no samples if no is negative (referred to as an advance).


2. ReversalThis transformation is given by f(n) = -n and simply involves "flipping" the signal x(n) with respect to the index n.


3. Time ScalingThis transformation is defined by f(n) = Mn or f(n) = n/N where M and N are positive integers. In the case of f(n) = Mn, the sequence x(Mn) is formed by taking every Mth sample of x(n) (this operation is known as down-sampling). With f (n) = n/N the sequence y(n) =x( f (n)) is defined as follows:

this operation is known as up-sampling.


Down-sampling by a factor of 2


Up-sampling by a factor of 2


Transformation of the Dependent Variable1. Addition

2. Multiplication

3. Scaling


Discrete-Time SystemsA discrete-time system is a mathematical operator or mapping that transforms one signal (the input) into another signal (the output) by means of a fixed set of rules or operations. The notation T[·] is used to represent a general system

where the symbol T denotes the transformation (also called an operator), or processing performed by the system on x(n) to produce y(n).


Block Diagram Representation of Discrete-Time Systems

1. Adder


2. Constant multiplier

3. Signal multiplier


4. Unit delay element

5. Unit advance element


Example:Using basic building block, sketch the block diagram representation of the discrete-time system described by the input-output relation


Classification of Discrete-Time Systems1. Static versus dynamic systems

A system is said to be static or memoryless if the output at any time n = n0 depends only on the input at time n = n0.


If the output of a system at time n is completely determined by the input samples in the interval from n – N to n (N ≥ 0), the system is said to have memory or dynamic of duration N.


2. Time-invariant versus time-variant systemsA system is called time-invariant if its input-output characteristics do not change in time.ifand

then

it is time-invariant if


A system is called time-variant if its input-output characteristics change in time.ifand

then

it is time-variant if


Example:Determine if the systems shown are time invariant or time variant.

(a)


The input-output equation is

The response of the system to x(n-k) is

if we delay y(n) by k units in time, we obtain

since y(n,k) = y(n – k), then the system is time invariant.


(b)


The input-output equation is

The response of the system to x(n-k) is

if we delay y(n) by k units in time, we obtain

since y(n,k) ≠ y(n – k), then the system is time-variant.


3. Linear versus nonlinear systemsA relaxed T system is linear if it satisfy the superposition principle and if and only if

If a relaxed system does not satisfy the superposition principle, it is called nonlinear.


The linear system must satisfy the two following conditions:

1. Additivity:

2. Homogeneity (or Scaling):

The two equations can combined into a single condition


Example:Determine if the systems described by the following input-output equations are linear or nonlinear.(a) y(n) = x(n2)(b) y(n) = x2(n)


(a) the response of the system to two separate input signals are

the response of the system to a linear combination of these two input signals is

a linear combination of the two outputs yields

since the two equations are identical, we conclude that the system is linear.



(b) the response of the system to two separate input signals are

andthe response of the system to a linear combination of these two input signals is

a linear combination of the two outputs yields

since the two equations are not equal, the system is nonlinear.

4. Causal versus noncausal systemsA system is said to be causal if the output of the system at any time depends only on present and past inputs but does not depend on future paths.

A system that has an output that depends not only on present and past inputs but also on future inputs is called noncausal.


5. Stable versus unstable systemsAn arbitrary relaxed system is said to be bounded input-bounded output (BIBO) stable if and only if every bounded input produces a bounded output.

for all n. If, for some bounded input sequence x(n), the output is unbounded (infinite), the system is classified as unstable.


Interconnection of Discrete-Time SystemsThere are two basic ways in which systems can be interconnected: in cascade (series) or in parallel.



y2(n) = T2[x(n)]

Response of a Discrete-Time LTI System A.Impulse Response:

The impulse response (or unit sample response) h[n] of a discrete-time LTI system (represented by T) is defined to be the response of the system when the input is δ[n], that is,


B. Response to an Arbitrary Input:The shifted unit impulse (or sample) sequence δ[n-k] is defined as


Any sequence x[n] can be expressed as

Since the system is linear, the response y[n] of the system to an arbitrary input x[n] can be expressed as


Since the system is time-invariant, we have

By substitution, we obtain

The equation indicates that a discrete-time LTI system is completely characterized by its impulse response h[n].


C. Convolution Sum:The previous equation defines the convolution of two sequences x[n] and h[n] denoted by

This equation is called the convolution sum. The output of any discrete-time LTI system is the convolution of the input x[n] with the impulse response h[n] of the system.


The process of computing the convolution between x[k] and h[k] involves the following four steps:

1. Folding. Fold h[k] about k = 0 to obtain h[-k].2. Shifting. Shift h[-k] by n to the right (or left) if n is

positive (or negative), to obtain h[n-k].3. Multiplication. Multiply x[k] by h[n-k] to obtain

the product sequence.4. Summation. Sum all the values of the product

sequence to obtain the value of the output at time n.


Example:The impulse response of a linear time-invariant system is

Determine the response of the system to the input signal


Fold h(k)


Multiply h(-k) to x(k)


Shift h(-k) to the right by one


Multiply h(1-k) to vo(k)


Fold h(1-k)


Multiply h(-1-k) to v1(k)


Any further shifts of h(-1-k) to the left always result in an all-zero product sequence, hence

The entire response of the system -∞ < n < ∞, which can be summarize as


D. Properties of the Convolution Sum:1. Commutative:


2. Associative:

3. Distributive:


Properties of Discrete-Time LTI SystemsA.Systems with or without Memory:

Since the output y[n] of a memoryless system depends on only the present input x[n], then, if the system is also linear and time- invariant, this relationship can only be of the

where K is a gain (constant). Thus, the corresponding impulse response is simply

therefore, if h[n0] ≠ 0 for n0 ≠ 0, the discrete-time LTI system has memory.


B. CausalityWe say that any sequence x[n] is called causal if

and is called anticausal if

Then when the input x[n] is causal, the output y[n] of a causal discrete-time LTI system isgiven by


C. StabilityAssume that the input x[n] of a discrete-time LTl system is bounded, that is,

Then we have,


o Therefore if the impulse response is absolutely summable, that is,

we have

and the system is BIBO stable


Systems with Finite-Duration and Infinite-Duration Impulse Response1.FIR

(causal)

(convolution)

FIR system has a finite memory of length M-samples


2. IIR

(convolution)

IIR system has an infinite memory


Eigenfunctions of Discrete-Time LTI SystemsThe eigenfunctions of discrete-time LTI systems

represented by T are the complex exponentials zn, with z a complex variable. That is,

where A is the eigenvalue of T associated with zn.


Setting x[n] = zn in convolution, we have

where


Thus the eigenvalue of a discrete-time LTI system associated with the eigenfunction zn is given by H(z) which is a complex constant whose value is determined by the value of z. Note that y[0] = H(z).


Systems described by Difference EquationsWithin the general class of IIR systems, this family of

discrete-time systems is more conveniently described by difference equations. This family or subclass of IIR systems is very useful in a variety of practical applications, including the implementation of digital filters, and the modeling of physical phenomena and physical systems.


A. Linear Constant-Coefficient Difference EquationsThe discrete-time counterpart of the general differential is the Nth-order linear constant-coefficient difference equation given by

where coefficient ak and bk are real constants. The Integer N is called the order of the difference equation or the order of the system. The order N refers to the largest delay of y[n].


A system is linear if it satisfies the following three requirements:

1. The total response is equal to the sum of the zero-input and zero-state responses.


2. The principle of superposition applies to the zero-state response (zero-state linear)

3. The principle of superposition applies to the zero-input response (zero-input linear)

A system that does not satisfy all three separate requirements is by definition nonlinear.


Example:Determine if the recursive system defined by the difference equation

is linear.


or, more compactly,


the expression can be expressed as

Thus the first requirement for linearity is satisfied.To check for the second requirement, let us assume that x(n) = c1x1(n) + c2x2(n). Then the zero-state response gives


Hence yzs(n) satisfies the principle of superposition, and thus the system is zero-state linear.Now let us assume that y(-1) = c1y1(-1) + c2y2(-1). From the zero-input response we obtain

Hence the system is zero-input linear.Since the system satisfies all three conditions for linearity, it is linear.


Solutions of Linear Constant-Coefficient Difference Equations

The homogeneous solution

The particular solution


B. Recursive FormulationRearranging the equation

in the form

we obtain a formula to compute the output at time n in terms of the present input and the previous values of the input and output.


There are many systems where it is necessary or desirable to express the output of the system not only in terms of the present and past values of the input, but also in terms of the already available past output values.


In general, a system whose output y(n) at time n depends on any number of past output values y(n – 1), y(n – 2), … is called a recursive system.


If y(n) depends only on the present and past inputs, then the system is called nonrecursive

(convolution sum for a causal FIR system).


From the equation we see that the need for auxiliary conditions is obvious and that to calculate y[n] starting at n = n0, we must be given the values of y[n0 – 1], y[n0 – 2], … , y[n0 - N] as well as the input x[n] for n ≥ n0 –M. The general form of the equation is called a recursive equation since it specifies a recursive procedure for determining the output in terms of the input and previous outputs.


In the special case when N = 0, we have

which is a nonrecursive equation since previous output values are not required to compute the present output. Thus, in this case, auxiliary conditions are not needed to determine y[n].


C. Impulse Responsethe impulse response h[n] of a discrete-time LTI system described by the linear constant-coefficient difference equation and recursive equation can be determined easily as


For the system described by the nonrecursive equation, the impulse response h[n] is given by

Note that the impulse response for this system has finite terms; Because of this property, the system specified by the equation is known as a finite impulse response (FIR) system. On the other hand, a system whose impulse response is nonzero for an infinite time duration is said to be an infinite impulse response (IIR) system.


Correlation of Discrete-Time SignalsTo measure the degree to which the two signals are similar and thus to extract some information that depends to a large extent on the application.Correlation of signals is often encountered in radar, sonar, digital communications, geology, and other areas in science and engineering.


Crosscorrelation and Autocorrelation SequencesThe crosscorrelation of x(n) and y(n) is a sequence rxy(l), which is defined as

or equivalently as


If we reverse the roles of x(n) and y(n) in the equations, we obtain the crosscorrelation sequence

or equivalently

by comparison, we conclude that


Example:Determine the crosscorrelation sequence rxy(l) of the sequences


For l = 0 we have

the product sequence vo(n) = x(n)y(n) is

and hence the sum over all values of n is


For l > 0, we simply shift y(n) to the right relative to x(n) by l units, compute the product sequence vl(n) = x(n)y(n – l), and finally, sum over all values of the product sequences. Thus we obtain


For l < 0, we simply shift y(n) to the left relative to x(n) by l units, compute the product sequence vl(n) = x(n)y(n – l), and finally, sum over all values of the product sequences. Thus we obtain

Therefore, the crosscorrelation sequence of x(n) and y(n) is


The convolution of x(n) and y(-n) yields the crosscorrelation rxy(l), that is,

In the special case where y(n) = x(n), we have the autocorrelation of x(n), which is defined as the sequence

or equivalently


For finite-duration sequences, the crosscorrelation and autocorrelation sequences may be expressed as

and

where i = l, k = 0 for l ≥ 0, and i = 0, k = l for l < 0


QUESTIONS


Discrete-time Signals and Systems

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