Discrete - time System and Analysis

Digital Signal Digital Signal ProcessingProcessing

Discrete-Time Signals Discrete-Time Signals and Systemsand Systems

Lecturer: Prof. Dr. M.J.E. Salami

Discrete-Time SignalsDiscrete-Time Signals

A discrete-time signal x(n) is a function of an independent variable that is an integer. It is assumed that a discrete-time signal is defined for every integer value n for - < n < . An example of a discrete-time signal is shown in the figure below.

Discrete-Time SignalsDiscrete-Time Signals

Besides the graphical representation of a discrete-time signal, there are some alternative representations that are often more convenient to use.

1. Functional representation, such as

2. Tabular representation, such as

3. Sequence representation

An infinite-duration signal or sequence with the time origin (n = 0) indicated by the symbol is represented as

n … -2 -1 0 1 2 3 4 5 …

x(n) … 0 0 0 1 4 1 0 0 …

Elementary Discrete-Time Elementary Discrete-Time SignalsSignals

Unit Sample Sequence

The unit sample sequence is denoted as (n) and defined as

In words, the unit sample sequence is a signal that is zero everywhere, except at n = 0 where its value is unity. The signal is sometimes referred to as unit impulse. The unit sample sequence is illustrated below.


Unit Step Signal

The unit step signal is denoted as u(n) and defined as

The unit step signal is illustrated below.


Unit Ramp Signal

The unit ramp signal is denoted as ur(n) and defined as

The unit ramp signal is illustrated below.


Exponential Signal

The exponential signal is a sequence of the form

If the parameter a is a real, then x(n) is a real signal. The figure below illustrates x(n) for various values of parameter a.


Exponential Signal

When the parameter a is complex valued, it can be expressed as

where r and θ are now parameters. Hence, x(n) can be expressed as

The signal x(n) can be represented graphically by the amplitude function

and the phase function


Graph of amplitude and phase function of a complex valued exponential signal

Classification of Discrete-Time Classification of Discrete-Time SignalsSignals

Energy signals and power signals:

The energy E of a signal x(n) is defined as

The energy of a signal can be finite or infinite. If E is finite (i.e., 0 < E < ), then x(n) is called an energy signal.

Many signals that possess infinite energy, have a finite average power. The average power of a discrete-time signal x(n) is defined as

If we define the signal energy of x(n) over the finite interval –N ≤ n ≤ N as


Then we can express the signal energy E as

And the average power of the signal x(n) as

Evidently, if E is finite, P = 0. On the other hand, if E is infinite, the average power P may be either finite or infinite. If P is finite (and nonzero), the signal is called a power signal.

ExampleExample

Determine the power and energy of the unit step sequence?


Periodic signals and aperiodic signals:A signal x(n) is periodic with period N( N > 0) if and only if

The smallest value of N for which the above equation holds is called the (fundamental) period. If there is no value of N that satisfies this equation, the signal is called nonperiodic or aperiodic.The energy of a periodic signal x(n) for 0 ≤ n ≤ N – 1, is finite if x(n) takes on finite value over the period. However, the energy of the periodic signal for - ≤ n ≤ is infinite. Whereas, the average power of the periodic signal is finite and it is equal to the average power over a single period. Thus if x(n) s a periodic signal with fundamental period N and takes on finite values, its power is given by

Consequently, periodic signals are power signals and aperiodic signals are energy signals.


Symmetric (even) and antisymmetric (odd) signals:

A signal x(n) is called symmetric (even) if

On the other hand, a signal x(n) is called antisymmetric (old) if

A diagrammatic illustration of even and odd signals is as shown below:


It can be illustrated that any signal can be expressed as the sum of two signal components, one of which is even and the other odd. The even signal component is formed by adding x(n) to x(-n) and dividing by 2, that is

Similarly, we form an odd signal component xo(n) according to the relation

It is hence evident that if we add the two signal components, we will obtain the composite signal x(n) which is as mathematically shown as

Manipulations of Discrete-Time Manipulations of Discrete-Time SignalsSignals

Transformation of the independent variable (time)

A signal x(n) can be shifted by replacing the independent variable n by n – k, where k is an integer. If k is a positive integer, the time shift results in a delay of the signal by k units of time. If k is a negative integer, the time shift results in advance of the signal by |k| units in time.

The figure below graphically illustrates a signal x(n).

ExampleExample

From the graphical illustration of signal x(n) given earlier, show a graphical representation of the signals x(n - 3) and x(n + 2).


Another modification of the time base is to replace the independent variable n by –n. The result of this operation is a folding or a reflection of the signal about the time origin n = 0.

It is important to note that the operations of folding and time delaying (or time advancing) a signal is not commutative. If we denote the time-delay operation by TD and the folding operation by FD, we can write

Note that because of the signs of n and k in x(n - k) and x(-n + k) are different, the result is a shift of the signals x(n) and x(-n) to the right by k samples, corresponding to a time delay.

ExampleExample

Show the graphical representation of the signal x(-n) and x(-n + 2), where x(n) is the signal illustrated in the figure below.


A third modification of the independent variable involves replacing n by µn, where µ is an integer. We refer to this time-base modification as time scaling or down-sampling.

If the signal x(n) was originally obtained by sampling an analog signal xa(t), then x(n) = xa(nT), where T is the sampling interval. Now, y(n) = x(2n) = xa(2Tn). Hence if the time-scaling operation is equivalet to changing the sampling rate from 1 / T to 1 / 2T, that is decreasing the rate by a factor of 2. This is a down-sampling operation.


Show the graphical representation of the signal y(n) = x(2n), where x(n) is the signal illustrated in the figure below.


Addition, multiplication, and scaling of sequences:

Amplitude modification include addition, multiplication, and scaling of discrete-time signals.

Amplitude scaling of a signal by a constant A is accomplished by multiplying the value of every signal sample by A. Consequently, we obtain

The sum of two signals x1(n) and x2(n) is a signal y(n), whose value at any instant is equal to the sum of the values of these two signals at that instant, that is,

The product of two signals is similarly defined on a sample-to-sample basis as

Manipulation of signalsManipulation of signals

Discrete-Time SystemsDiscrete-Time Systems

A discrete-time system is a device or algorithm that operates on a discrete-time signal, called the input or excitation, according to some well-defined rule, to produce another discrete-time signal called the output or response of the system.

We say that the input signal x(n) is transformed by the system into a signal y(n), and express the general relationship between x(n) and y(n) as

Where the symbol T denotes the transformation (also called an operator).


The mathematical relationship is depicted graphically as a block diagram representation of a discrete-time system as shown below.

ExampleExample


The Accumulator:

It is evident that the output at time n = n0 depends not only on the input at time n = n0, but also on x(n) at times n = n0 – 1, n0 – 2, and so on. By a simple algebraic manipulation, the input-output relation of the accumulator can be written as

which justifies the term accumulator.

Suppose that we are given the input signal x(n) for n ≥ n0, and we wish to determine the output y(n) of this system for n ≥ n0. For n = n0, n0 + 1,…, the equation above yields


It should be noted there is a problem in computing y(n0),since it depends on y(n0 - 1). However,

Thus the response of the system for n ≥ n0 to the input x(n) that is applied at time n0 is the combined result of this input and all inputs that had been applied previously to the system. Consequently, y(n), n ≥ n0 is not uniquely determined by the input x(n) for n ≥ n0.

The additional information required to determine y(n) for n ≥ n0 is the initial condition y(n0 - 1). This value summarizes the effect of all previous inputs to the system.

If the accumulator had no excitation prior to n0, the initial condition is y(n0 - 1) = 0. In such a case we say that the system is initially relaxed.

It is customary to assume that every system is relaxed at n = -. In this case, if an input x(n) is applied at n = -, the corresponding output y(n) is solely and uniquely determined by the given input.

ExampleExample

The accumulator (of question 6 of the previous example) is excited by the sequence x(n) = nu(n). Determine its output under the condition that:

(a) It is initially relaxed [i.e., y(-1) = 0].

(b) Initially, y(-1) = 1.

Block Diagram Representation of Block Diagram Representation of Discrete-Time SystemsDiscrete-Time Systems

An adder

The adder is a system that performs the addition of two signal sequences to form another (the sum) sequence, which we denote as y(n) as shown below. Note that it is not necessary to store either one of the sequences in order to perform the addition. In other words, the addition operation is memoryless.

+


A Constant Multiplier

This operation depicted in the figure below. It represent applying a scale factor on the input x(n). This operation is also memoryless.


A Signal Multiplier

The figure below illustrates the multiplication of two signal sequences to form another (the product) sequence, denoted as y(n). The multiplication operation is also memoryless.


A Unit Delay Element

The unit delay is a special system that simply delays the signal passing through it by one sample. If the input of the system is x(n), the output is x(n – 1). The sample x(n–1) is stored in memory at time n – 1 and it is recalled from memory at time n to form

The figure below illustrates the unit delay element.

z-1


A Unit Advance Element

In contrast to the unit delay, a unit advance moves the input x(n) ahead by one sample in time to yield x(n + 1). The figure below illustrates the unit advance element.

z

ExampleExample

Using the basic building blocks introduced, sketch the block diagram representation of the discrete-time system described by the input-output relation.

where x(n) is the input and y(n) is the output.

Classification of Discrete-Time Classification of Discrete-Time SystemsSystems

Static versus dynamic systems:

A discrete-time system is called static or memoryless if its outptut at any instant n depends at most on the input sample at the same time, but not on past or future samples of the input. In any other case, the system is said to be dynamic or to have memory.

Note:

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 of duration N.

If N = 0, the system is static.

If 0 < N < , the system is said to have finite memory.

If N = , the system is said to have infinite memory.

ExamplesExamples


Time-invariant versus time-variant systems:

A system is called time-invariant if its input-output characteristics do not change with time. The output signal y(n) from a system T excited by an input signal x(n) can be denoted as

Definition: A relaxed system T is time invariant or shift invariant if and only if

Implies that

for every input signal x(n) and every time shift k.

ExamplesExamples

+

z-1

Differentiator-

“Time” multiplier

+

Modulator

T“Folder”

Determine and justify whether the following systems are time-invariant or time-variant


Linear versus nonlinear systems:

A linear system is one that satisfies the superposition principle.

Superposition Principle:

This principle requires that the response of the system to a weighted sum of signals be equal to the corresponding weighted sum of the responses (outputs) of the system to each of the individual input signals.

Definition:

A relaxed T system is linear if and only if


Graphical representation of the superposition principle:

+ T

a1

a2

x1(n)

x2(n)

y(n)

T

T

+

x1(n)

x2(n)

a1

a2

y’(n)

T is linear if and only if y(n) = y’(n)

ExamplesExamples


Causal versus noncausal systems:

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 inputs [i.e., x(n), x(n - 1), x(n - 2),…], but does not depend on future inputs [i.e., x(n + 1), x(n + 2),…].

If a system does not satisfy this definition, it is called noncausal.

ExamplesExamples


Stable versus unstable systems:

An arbitrary relaxed system is said to be bounded input-bounded output (BIBO) stable if and only if every bounded input produces a bounded output.

ExampleExample

Interconnection of Discrete-Time Interconnection of Discrete-Time SystemsSystems

Discrete-time systems can be interconnected to form larger systems in two basic ways: cascaded (series) and parallel. These interconnections are illustrated below:

T1 T2

T1

T2

+

x(n) y1(n) y(n)

x(n)

y1(n)

y2(n)

y3(n)

Tc

Tp

Cascade Interconnections of

systems

Parallel Interconnections of

systems

Techniques for the Analysis of Techniques for the Analysis of Linear SystemsLinear Systems

There are two basic methods fir analyzing the behavior or response of a linear system to a given input signal. One method s based on the direct solution of the input-output equation for the system, which in general has the form

where F[·] denotes some functions of the quantities in brackets. The general form of the input-output relationship for an LTI system is

where {ak} and {bk} are constant parameters that specify the system and are dependent of x(n) and y(n).


The second method for analyzing the behavior of a linear system to a given input signal is to first decompose or resolve the input signal into sum of elementary signals. The elementary signals are selected so that the response of the system to each signal component is easily determined. Then using the linearity property of the system, the responses of the system to the elementary signals are added to obtain the total response of the system to the given input signal.

To elaborate this method further, suppose that the input signal x(n) is resolved a into weighted sum of elementary signal components {xk(n)} so that

where the {ck} are the set of amplitudes (weighing coefficients) in the decomposition of the signal x(n).


Now, suppose that the response of the system to the elementary signal component xk(n) is yk(n). Thus,

assuming that the system is relaxed and that the response to ckxk(n) is ckyk(n), as a consequence of scaling property of the linear system.

Finally, the total response to the input x(n) is

Resolution of a Discrete-Time Resolution of a Discrete-Time Signal into ImpulsesSignal into Impulses

Suppose we have an arbitrary signal x(n) that we wish to resolve the sum into a sum of sample sequences. We select the elementary signals xk(n) to be

where k represents the delay of the unit sample sequence. To handle an arbitrary signal x(n) that must have nonzero values over an infinite duration, the set of unit impulses must also be infinite, to encompass the infinite number of delays.

Now suppose that we multiply the two sequences x(n) and (n – k). Since (n – k) is zero everywhere except at n = k, where its value is unity, the result of this multiplication is another sequence that is zero everywhere except at n = k, where its value is x(k). Thus

Is a sequence that is zero everywhere except at n = k, where its value is x(k). If we repeat the multiplication of x(n) and (n – m), where m is another delay (m ≠ k), the result will be a sequence that is zero everywhere except at n = m, where it is x(m).

Resolution of a Discrete-Time Resolution of a Discrete-Time Signal into ImpulsesSignal into Impulses

In other words, each multiplication of the signal x(n) by a unit impulse at some delay k, [i.e. (n – k)], in essence picks out the single value x(k) of the signal x(n) at the delay where the unit impulse is nonzero. Consequently, if we repeat this multiplication over all possible delays, - < k < , and sum all the product sequences, the result will be a sequence x(n), that is,

Example:

Consider the special case of a finite-duration sequence given as

Resolve the sequence x(n) into a sum of weighted impuluse sequences.

Response of LTI Systems to Arbitrary Response of LTI Systems to Arbitrary Inputs: The Convolution SumInputs: The Convolution Sum

To determine the response of any relaxed linear system to any input signal, we denote the response y(n, k) of the system to the input unit sample sequence at n = k by the special symbol h(n, k), - < k < . That is,

If the impulse at the input is scaled by an amount ck ≡ x(k), the response of the system is the correspondingly scaled output, that is,

Finally, if the input is the arbitrary signal x(n) is corresponding sum of weighted outputs that is

Superposition summation


If the response of the LTI system to the unit sample sequence (n) is denoted as h(n), that is,

then by the time-invariance property, the response of the system to the delayed unit sample sequence (n – k) is

Consequently, the superposition summation formula reduces to

The formula above is called a convolution sum

The process of computing the convolution between x(k) and h(k) involves the following four steps.

1. Folding. Fold h(n) about k = 0 to obtain h(-k)

2. Shifting. Shift h(-k) by n0 to the right (left) if n0 is positive (negative), to obtain h(n0 – k).

3. Multiplication. Multiply x(k) by h(n0 – k) to obtain the product sequence vno(k) ≡ x(k)h(n0 – k).

4. Summation. Sum all the values of the product sequence vno(k) to obtain the value of the output at the time n = n0.


ExampleExample

1. The impulse response of a linear time-invariant system is

Determine the response of the system to the input signal

2. Determine the output y(n) of a relaxed linear time-invariant system with impulse response

When the input is a unit step sequence, that is

Properties of Convolution and the Properties of Convolution and the Interconnection of LTI SystemsInterconnection of LTI Systems

Properties of Convolution and the Properties of Convolution and the Interconnection of LTI SystemsInterconnection of LTI Systems

Commutative Law:

As shown earlier the convolution property can be mathematically expressed as:

Associative Law:

In mathematical terms, it can be shown that the convolution operation also satisfies the associative property as shown below:

Distributive Law:

It can also be shown through mathematical means that the convolution operation also satisfies the distributive property as shown below:

ExampleExample

ExampleExample

ExampleExample

ExampleExample

Recursive and Nonrecursive Recursive and Nonrecursive Discrete-Time SystemsDiscrete-Time Systems

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. An example of a recursive system is the cumulative averaging system represented as

+

z-1

Realization of a recursive cumulative averaging system

Recursive and Nonrecursive Recursive and Nonrecursive Discrete-Time SystemsDiscrete-Time Systems

The output of a causal and practically realizable recursive system can be expressed in general as

If the output of a system y(n) depends only on the present and past inputs, then

Such a system is called nonrecursive.

The figure below reveals the basic difference between a recursive system and a nonrecursive system.

F[x(n), x(n–1), … x(n–M)]

x(n) y(n)

Nonrecursive system

F[y(n–1), .. y(n–1), x(n), x(n–1), … x(n–

M)]

x(n) y(n)

z-1

Recursive system

Linear Time-Invariant Systems Characterized Linear Time-Invariant Systems Characterized by Constant-Coefficient Difference Equationsby Constant-Coefficient Difference Equations

A recursive system is relaxed if it starts with zero initial condition. Because the memory of a system describes, in some sense, its “state,” we say that the system is at zero state and its corresponding output is called the zero-state response or forced response and is denoted by yzs(n).

If the system is initially nonrelaxed and the input x(n) = 0 for all n, the output of the system with zero input is called zero-input response and is denoted by yzi(n).

The zero-input response is obtained by setting the input signal to zero, making it independent of the input. IT depends only on the nature of the system and the initial condition. On the other hand, the zero-state response depends on the nature of the system and the input signal.

In general, the total response of the system is expressed as

Linear Time-Invariant Systems Characterized Linear Time-Invariant Systems Characterized by Constant-Coefficient Difference Equationsby Constant-Coefficient Difference Equations

A system that is linear 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)

Example

Determine if the recursive system defined by the difference equation

is linear.

Solution of Linear Constant-Solution of Linear Constant-Coefficient Difference EquationsCoefficient Difference Equations

Homogeneous SolutionHomogeneous Solution

ExampleExample

ExampleExample

Particular SolutionParticular Solution

ExampleExample

ExampleExample

Total SolutionTotal Solution

ExampleExample

ExampleExample

The Impulse Response of a Linear The Impulse Response of a Linear Time-Invariant Recursive SystemTime-Invariant Recursive System

The impulse response of a linear time-invariant recursive system, h(n) is equal to the zero-state response of the system when the input x(n) = (n) and the system is initially relaxed.

In the general case of an arbitrary linear-time invariant recursive system, the zero-state response expressed in terms of the convolution summation is

When the input is an impulse [i.e. x(n) = (n)], the above equation reduces to

ExampleExample

Structures for the Realization of Structures for the Realization of Linear Time-Invariant SystemsLinear Time-Invariant Systems

The realization for the first order system

Is shown below. This realization is called a direct form I structure because it uses separate delays (memory) for both the input and output signal samples. This system can be viewed as two linear time-invariant systems in cascade described by the equation.

+ +

z-1 z-1


An alternative structure for the realization or the system is shown below which results in the following difference equations

This structure can be reconstructed by merging the delays in the structure. This new structure is called the direct form II structure.

+ +

z-1

Direct form II realization

+ +

z-1 z-1

Alternate Direct form I realization


The structures for realizing a second-order system is shown below

+

+ +

+

z-1

z-1

z-1 z-1

+ +

+ +

z-1 z-1

Further ExampleFurther Example


Further ExamplesFurther Examples

Further ExamplesFurther Examples

Example of the use of Correlation of SignalsExample of the use of Correlation of Signals

ExampleExample

Correlation of Discrete-Time SignalsCorrelation of Discrete-Time SignalsCrosscorrelation Sequences

ExampleExample

Correlation of Discrete-Time SignalsCorrelation of Discrete-Time Signals

Autocorrelation Sequences

Properties of the Autocorrelation Properties of the Autocorrelation and Crosscorrelation Sequencesand Crosscorrelation Sequences

Assume we have two sequences x(n) and y(n) with finite energy from which we form the linear combination,

It can be derived that the crosscorelation sequence satisfies the condition that

In a special case where x(n) = y(n), this reduces to

This means that the autocorrelation sequence of a signal attains its maximum value at zero lag.

Properties of the Autocorrelation Properties of the Autocorrelation and Crosscorrelation Sequencesand Crosscorrelation Sequences

The normalized autocorrelation sequence is defined as

While the normalized crosscorrelation sequence is defined as

The autocorrelation sequence also satisfies the property

ExampleExample

Compute the autocorrelation of the signal



Correlation of Periodic SequencesCorrelation of Periodic Sequences

ExampleExample

Example contdExample contd

ExampleExample



Computation of Correlation Computation of Correlation SequencesSequences

Input-Output Correlation Input-Output Correlation SequencesSequences

Assume that a signal x(n) with known autocorrelation rxx(l) is applied to an LTI system with impulse response h(n), producing the output signal

The crosscorrelation between the output and input signal is

The autocorrelation of the output signal can be obtained using the properties of convolution as

Random SignalsRandom Signals





ExamplesExamples

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Discrete - time System and Analysis