DSP Lectures v2 [Chapter2][1]

Signals, Spectra and Signal Processing (EC413L1) Chapter 2 DISCRETE-TIME SIGNALS AND SYSTEMS 2.0 Introduction •

• •

This chapter introduces the different elementary discrete-time signals that are important in the treatment of signal processing. These are used as basis functions or building blocks to describe more complex signals. This chapter also emphasizes the characterization of discrete-time systems in general and the class of linear time-invariant (LTI) systems in particular. The motivation for studying LTI systems is twofold: first, there are a large collection of mathematical techniques that can be applied to the analysis of LTI systems; second, many practical systems are either LTI systems or can be approximated by LTI systems.

2.1 Discrete-Time Signals •

A discrete-time signal x(n) is a function of an independent variable that is an integer. A graphical representation of a discrete-time signal is shown below Discrete-Time Signal 2

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x(n)

1

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-0.5

-1 -4

-3

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-1

0

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n

Figure 2.1. Graphical representation of discrete-time signal

Note: A discrete-time signal is NOT DEFINED at instants between two successive samples. • •

A discrete-time signal is defined for every integer value n for -∞ < n < +∞. x(n) refers to the nth sample of the signal.

Chapter 2 – Discrete-Time Signals and Systems

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Some alternative representations of discrete-time signals: o Functional representation – through equations: Example: 1, for n = 1,3 xn =  4, for n = 2  0, elsewhere

o

Tabular representation – through tables: Example:

o

Sequence representation – through row matrices: Examples: For infinite-duration signal, the time origin (n = 0) is indicated by an ↑below the value.  x (n ) = . . . 0 

0 ↑

1

4

1

0

 0. . .  

A sequence, which is zero for n < 0 can be represented as   x (n ) = 0 1 4 1 0 0 . . .  ↑ 

If an arrow is omitted, the leftmost value is understood to be the sample at time-origin. A finite-duration sequence is represented as   x (n ) = 3 − 1 − 2 5 0 4 − 1 ↑  

whereas a finite-duration sequence that is defined only at n > 0 can be represented as x (n ) = [0 1 4 1]

2.1.1 Some Elementary Discrete-Time Signals • Unit sample sequence, denoted as δ(n), is defined as

1 for n = 0 δn =  0 for n ≠ 0

2.1.6

In words, the unit sample sequence is a signal that is zero everywhere, except at n = 0 where its value is unity. It is also called unit impulse. The graphical representation of δ(n) is shown below.

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Figure 2.2 Unit impulse sequence

•

Unit step signal, denoted as u(n), is defined as

1 for n ≥ 0 un =  0 for n < 0

2.1.7

Unit step signal is illustrated below 2

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Figure 2.3. Unit step sequence

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Signals, Spectra and Signal Processing (EC413L1) •

Unit ramp signal, denoted as r(n), and is defined as

n for n ≥ 0 rn =  0 for n < 0

2.1.8

Unit ramp sequence is shown below 15

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-5

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Figure 2.4. Unit ramp sequence

•

Exponential signals are of the form

xn = a

2.1.9

If a is a real number, then x(n) is also a real number. The figures below illustrate x(n) for various values of the parameter a.

Figure 2.5. Real exponential signals for various values of the parameter a

Chapter 2 – Discrete-Time Signals and Systems

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Signals, Spectra and Signal Processing (EC413L1) When the parameter a is complex-valued, it can be expressed as

and x(n) as

a = re

xn = !re " = r  ∙ e 

having a real part xR(n)

and imaginary part xI(n)



= r  cos θn + j sin θn

2.1.10

xR n = r  cos θn

2.1.11

xI n = r  sin θn

2.1.12

Figure below shows the real and imaginary plots of the complex exponential signal. Notice that the real part is a damped cosine wave while the imaginary part is a damped sine wave. real part 1 0.5 0 -0.5

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imaginary part 1 0.5 0 -0.5

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Figure 2.6. Graphs of real and imaginary components of a complex-valued signal

Complex exponential signals can also be represented graphically using the amplitude function

and the phase function

|xn| = An = r 

2.1.13

∠xn = ϕn = θn

2.1.14

The figure below shows the plot of complex exponential function in terms of magnitude and phase functions.

Chapter 2 – Discrete-Time Signals and Systems

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phase 5 0 -5

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Figure 2.7. Graph of amplitude and phase function of a complex-valued exponential signal

2.1.2 Classification of Discrete-Time Signals •

Energy and power signals o The energy E of a signal x(n) is given as 2

E = 0 |xn|1

2.1.15

3 42

o o

If the energy of the signal E is finite, then it is an energy signal. If its energy is infinite, it may have finite average power, given by N

1 0 |xn|1 P = lim N → 2 2N + 1

2.1.16

3 4N

o •

If it has finite average power, then it is a power signal.

Periodic and aperiodic signals o A signal x(n) is periodic with period N (N > 0) if and only if xn + N = xn for all n

o

2.1.20

If there is no value of N that satisfies the above equation, the signal is considered aperiodic.

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Signals, Spectra and Signal Processing (EC413L1) o

o

The energy of a periodic signal x(n) over a single period is finite, if x(n) takes on finite values over the period. However, for the whole duration of the signal (from negative to positive infinity) its energy is infinite. For the whole duration of the signal, the average power of the periodic signal is finite and is equal to the average power over a single period, given by N4:

1 0|xn|1 P= N

2.1.23

3;

hence, a periodic signal is a power signal. •

Example: Determine whether the following signals are energy or power signals and determine their energy and power. a) Unit sample sequence b) Unit step sequence c) xn = cos < ?, for −∞ < n < +∞ > =

•

d) xn = 0.2 un   e) x (n) = L 3 − 2 1 − 1 4 3 2 L ↑   Symmetric (even) and antisymmetric (odd) signals o A real-valued signal x(n) is called symmetric or even if x(-n) = x(n) o It is asymmetric or odd if x(-n) = -x(n). We note that in this case, x(0) = 0. Even 4 2 0 -5

0

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Odd 5 0 -5 -5

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Figure 2.8. Graphical illustration of even and odd signals

o

Any signal can be expressed as the sum of two signal components, one even and one odd.

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The even signal component is formed by xC n =

o

2.1.26

The odd signal component is formed by xF n =

•

1 Dxn + x−nE 2 1 Dxn − x−nE 2

2.1.27

Example: Resolve the following signals into its odd and even components and plot the resulting sequence. Verify your answers. a) Unit step sequence  1 + > for − 3 ≤ n ≤ −1  b) xn = G 1 for 0 ≤ n ≤ 3 c)

0 elsewhere

  x (n) = L 3 − 2 1 − 1 4 3 2 L ↑  

2.1.3 Simple Manipulations of Discrete-Time Signals

•

Transformation of the independent variable (time) o Time shifting – shifting the signal x(n) in time involves replacing n with n – k, where k is an integer. If k is positive, the time shifts results in the delay of the signal by k units of time. If k is negative, the time shifts results in an advance of the signal by |k| units of time.

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Figure 2.9. Graphical representation of a signal, and its delayed and advanced versions

o

Time folding – the signal x(n) is folded about n = 0 when the time variable n is replaced by –n.

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Figure 2.10. Graphical illustration of the folding and shifting operations

o

o

Time-scaling – the signal x(n) is downsampled when the time variable n is replaced by an, where a is an integer. It is upsampled when the time variable n is replaced by n/a, where a is an integer. Downsampling means resampling the sampled signals, that is, decreasing the sampling rate by a. Upsampling means inserting a – 1 samples in between samples.

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Figure 2.11. Downsampling and Upsampling

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Addition, multiplication and scaling of sequences o Amplitude scaling of a signal by a constant A is accomplished by multiplying the value of every signal sample by A. yn = Axn for -∞ < n < +∞

o

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 yn = x: n + x1 n for -∞ < n < +∞

o

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The product of two signals is similarly defined on a sample-to-sample basis as yn = x: n ⋅ x1 n for -∞ < n < +∞

Example: A discrete-time signal is defined as xn = K

1+

n for − 3 ≤ n ≤ −1 3  1 for 0 ≤ n ≤ 3 0 elsewhere

a. Determine its values and sketch the signal x(n). b. Sketch the signal that will result if we: i. First fold x(n) and then delay the resulting signals by four samples. ii. First delay x(n) by four samples and then fold the resulting signal. Chapter 2 – Discrete-Time Signals and Systems

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Signals, Spectra and Signal Processing (EC413L1) c. Sketch the signal x(-n + 4). d. Compare the results in parts (b) and (c) and derive a rule for obtaining the signal x(-n+k) from x(n). e. Express the signal x(n) in terms of signal of unit sample and unit step sequences.

•

Example: A discrete-time signal x(n) is shown below. Sketch and label carefully each of the following signals:

a) b) c) d) e) f) g) h)

x(n – 2) x(4 – n) 3x(n + 2) x(n) u(2 – n) x(n – 1) δ(n – 3) x(n2) even part of x(n) odd part of x(n)

Chapter 2 – Discrete-Time Signals and Systems

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Signals, Spectra and Signal Processing (EC413L1) 2.2 Discrete-Time Systems

• • •

A discrete-time system is a device or an algorithm that performs some prescribed operation on a discrete-time signal. The discrete-time system performs operation on an input discrete-time signal according to some well-defined rule (the algorithm) to produce the output or response. We can describe the operation applied by the system to the input signal x(n) to produce the output y(n) in the following manner yn = ΤDxnE

2.2.1

where T denotes transformation (also called an operator) or processing performed by the system on x(n) to produce y(n). 2.2.1 Input – Output Description of Systems

•

The input – output description of a discrete-time system consists of a mathematical expression or a rule, which explicitly defines the relation between the input and output signals

Figure 2.12. Block diagram representation of a discrete-time system.

•

Example: For the input signal

|n|, xn = M 0,

−3 ≤ N ≤ 3 OPℎRSTUVR

Determine the output of the system defined by the following input – output relationship. a) yn = xn b) yn = xn − 1 c) yn = xn + 1 : d) yn = > Dxn + 1 + xn + xn − 1E

e) yn = maxDxn + 1, xn, xn − 1E f) yn = ∑Y3 42 xn = xn + xn − 1 + xn − 2 + ⋯ Chapter 2 – Discrete-Time Signals and Systems

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Signals, Spectra and Signal Processing (EC413L1) 2.2.2 Block Diagram Representation of Discrete-Time Systems

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An adder – a system that performs the addition of two signals sequences to form another sequence, which is the sum of the two inputs. The symbol for an adder is illustrated below

Figure 2.13. Symbol for adder

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A constant multiplier – a system that multiplies the input signal by a scale factor. The symbol for the constant multiplier is shown below.

Figure 2.14. Symbol for constant multiplier

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Signal multiplier – a system that multiplies two input sequences to produce the product sequence. A graphical representation of the signal multiplier appears below.

Figure 2.15. Symbol for signal multiplier

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Signals, Spectra and Signal Processing (EC413L1) •

Unit delay element – a system that simply delays the signal passing through it by one sample. The figure below illustrates such system.

Figure 2.16. Symbol for unit delay element

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Unit advance element – a system that moves the input ahead by one sample. A unit advance system is graphically illustrated as the figure below.

Figure 2.17. Symbol for unit advance element

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Example: Sketch the block diagram representation of the discrete-time system described by the input-output relation yn =

1 1 1 yn − 1 + xn + xn − 1 4 2 2

2.2.3 Classification of Discrete-Time Systems

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Static versus dynamic systems o A discrete-time system is called static or memoryless if its output at any instant depends at most on the input sample at the same time, but not on past or future samples of the input. The systems described by the following input-output equations yn = axn

yn = nxn + bx > n

are both static or memoryless. Note that there is no need to store any of the past inputs or ouputs in order to compute the present output.

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Signals, Spectra and Signal Processing (EC413L1) o

If the output of the system at any instant is computed using the past or future sample of the input and the past output, the system is said to be dynamic or to have memory. The systems described by the following input-output relations yn = xn + 3xn − 1 

yn = 0 xn − k Y3; 2

yn = 0 xn − k Y3;

are dynamic systems or systems with memory.

•

Time-invariant versus time-variant systems o A system is time-invariant if its input-output characteristics do not change with time. If a system T in a relaxed state is excited by the input x(n) to produce the output y(n), then we have yn = TDxnE

2.2.13

If we excite the system by an input delayed by k number of samples, the output becomes yn, k = TDxn − kE

The system is said to be time-invariant if y(n,k) = y(n – k); otherwise it is said to be time varying. o

The system identified by the input-output equation yn = xn − xn − 1

is time-invariant while the system

is time-varying system.

•

yn = nxn

Example: Determine if the following systems are time-invariant or time-varying systems a) yn = xn − xn − 1 b) yn = nxn c) yn = x−n d) yn = xn cos ω; n

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Linear versus nonlinear systems o A linear system is one that satisfies the superposition principle. o The superposition principle requires that the response of the system to a weighted sum of signals be equal to the corresponding weighted sum of the responses of the system to each of the individual input signals. o A relaxed system T is linear if and only if TDa: x: n + a1 x1 nE = a: TDx: nE + a1 TDx1 nE

2.2.26

for any arbitrary input sequences x1(n) and x2(n) and any arbitrary constants a1 and a2.

Figure 2.20. Graphical representation of the superposition principle

Linear systems exhibit multiplicative or scaling property and additive property. This is the consequence of the definition of the superposition principle (Eq. 2.2.26) o The linearity condition stated by Eq. 2.2.26 can also be extended to any weighted linear combination of signals. Example: Determine if the following systems described by the following input-output equations are linear or nonlinear. a) yn = nxn b) yn = xn1  c) yn = x 1 n d) yn = Axn + B e) yn = e_ o

•

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•

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Causal versus noncausal systems o A system is said to be causal if the output of the system at any time [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), …]. o If a system does not satisfy this condition, then it is said to be noncausal. o Noncausal systems are physically unrealizable in real-time signal processing applications. Example: Determine if the systems described by the following input-output equations are causal or noncausal o yn = xn − xn − 1 o yn = ∑Y3 42 xk o yn = axn o yn = xn + 3xn + 4 o yn = xn1  o yn = x2n o yn = x−n Stable versus unstable systems o Stability is an important property that must be considered in any practical application of a system o An arbitrary relaxed system is said to be bounded input – bounded output (BIBO) stable if and only if every bounded (finite) input produces a bounded output. o If, for some bounded input sequence x(n), the output is unbounded (infinite), the system is classified as unstable. Example: Analyze the stability of the nonlinear system described by the input-output equation yn = y 1 n − 1 + xn

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Signals, Spectra and Signal Processing (EC413L1) 2.3 Analysis of Discrete-Time Linear Time-Invariant Systems

• • • •

We now turn our attention to the analysis of the important class of linear, time-invariant (LTI) systems. In particular, we shall demonstrate that such systems are characterized in the time domain simply by their response to a unit sample sequence. We shall also demonstrate that any arbitrary input signal can be decomposed and represented as a weighted sum of unit sample sequences. The general form of the expression that relates the unit sample response of the system and the arbitrary input signal to the output signal, called the convolution sum or convolution formula, is also derived.

2.3.1 Techniques for the Analysis of Linear Systems

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Two basic methods for analyzing the behavior or response of a linear system to a given input signal. o Direct solution of the input-output relationship of the system, whose general form (for the LTI discrete-time systems) is N

M

yn = − 0 aY yn − k + 0 bY xn − k Y3:

2.3.1

Y3;

called the difference equation and represents one way to characterize the behavior of a discrete-time LTI systems.

• •

o

The second method of analyzing the behavior of a linear system to a given input signal is first decompose or resolve the input signal into a 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.

o

The first method is discussed in the next section, the second method is discussed in this section.

The choice of the elementary signals appears to be arbitrary, as long as the response can be determined conveniently. Resolution of the input signal to a weighted sum of unit sample (impulse) sequence proves to be mathematically convenient and completely general solution to the response of the system. But if the input signal is periodic with period N, it can be more mathematically convenient for us to resolve these signals into harmonically related exponentials xY n = eab 

Chapter 2 – Discrete-Time Signals and Systems

k= 0, 1, 2. . . N-1

2.3.5 Page 19

Signals, Spectra and Signal Processing (EC413L1) where the frequencies ωk are harmonically related and equal to 2πk/N. 2.3.2 Resolution of Discrete-Time Signals into Impulses

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Suppose we have an arbitrary signal x(n) that we wish to resolve into a sum of unit sample sequence. We select the elementary signals xk(n) to be xY n = δn − k

2.3.7

Note that the signal δ(n – k) is zero everywhere except at n = k, where its value is unity.

•

If we multiply the input signal x(n) with δ(n – k), the result of this multiplication is another sequence that is zero everywhere except at n = k, where its value is x(k). Thus if we repeat this process at all possible values of k, the equation below holds true: 2

xn = 0 xk δn − k

2.3.10

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  Example: Resolve the sequence x (n) = 2 4 0 3 into a sum of weighted impulse  ↑  sequence.

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

•

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We denote the response of the system to the input unit sample sequence as h(n) and is called the impulse response of the system. A relaxed LTI discrete-time system is characterized, in timedomain, by its impulse response. The response of the LTI system as a function of the input signal x(n) and its impulse response h(n) is given as 2

yn = 0 xnhn − k

2.3.17

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and is called the convolution sum of x(n) and h(n).

•

•

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The convolution sum involves four steps. o Folding – Fold h(k) about k = 0 to obtain h(-k). o Shifting – Shift h(-k) by n0 to the right (or left) if n0 is positive (or negative) to obtain h(n0 – k). o Multiplication – Multiply x(k) by h(n0 – k) to obtain the product sequence x(k)h(n0 – k). o Summation – Sum all the values of the product sequence to obtain the value of the output at time n = n0. We note that this procedure results in the response of the system at a single time instant n = n0. To evaluate the response of the system over all time instants, we should repeat steps 2 to 4 accordingly. We also note that the convolution sum is commutative, that is, it can be also expressed in the form 2

yn = 0 xn − khn

•

2.3.28

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

  h(n) = 1 2 1 − 1  ↑  Determine the response of the system to the input signal   x (n ) = 1 2 3 1  

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Example: Determine the output y(n) of a relaxed linear time-invariant system with impulse response hn = a un, |a| < 1

when the input is a unit step sequence, that is,

xn = un

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Example: Compute the convolution sum of the following pairs of signals

•

Example: Compute the convolution sum of the following pair of signals

2.3.4 Properties of Convolution and the Interconnection of LTI Systems

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To simplify the notation of convolution, the following forms are used:

and

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It was earlier noted that the convolution sum is commutative, that is

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The convolution sum is also associative, that is

xn ∗ hn = hn ∗ xn

Dxn ∗ h: nE ∗ h1 n = xn ∗ Dh: n ∗ h1 nE

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The distributive property of the convolution sum is described as follows:

xn ∗ Dh: n + h1 nE = xn ∗ h: n + xn ∗ h1 n

2.3.5 Causal Linear Time-Invariant Systems

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A causal system is one whose output at time n depends only on present and past inputs but does not depend on future inputs. For LTI systems, the causality condition also puts a restriction on the impulse response of the system. A causal system has a causal impulse response, that is h(n) = 0 for n < 0. It is a necessary and sufficient condition for causality.

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The convolution sum formula can now be modified to reflect this condition. Thus

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If the input signal x(n) is also causal, that is x(n) = 0 for n < 0, the above formula can be further simplified, that is

2.3.6 Stability of Linear Time-Invariant Systems

• •

•

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An arbitrary relaxed system is BIBO stable if and only if its output sequence y(n) is bounded for every bounded x(n). In terms of its impulse response h(n), an LTI system is stable if and only if its impulse response is absolutely summable, that is, it decays over time, or

This implies that any excitation at the input of the system, which is of finite duration, produces an output that is “transient” in nature, that is its amplitude decays with time and dies out eventually, when the system is stable. Example: Determine the range of values of the parameter a for which the LTI system with impulse response h(n) = an u(n) is stable.

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Example: Determine the range of values of a and b for which the LTI system whose impulse response

is stable.

a , hn = M  b ,

n ≥ 0 n