Systems described by difference equations

For the linear time-invariant discrete-time case, the role of differential equations is taken over by the so-called difference equations of the type

b₀y[n]+ b1y[n− 1] + · · · + bMy[n− M]

= a0u[n]+ a1u[n− 1] + · · · + aN[n− N].

This equation for the input u[n] and the output y[n] is called a linear difference equation with constant coefficients. The systems described by difference equations

are of major importance for the practical realization of systems. These will be dis-cussed in detail in chapter 19.

EXERCISES

For a continuous-time system the response y(t) to an input u(t) is given by 1.9

y(t) =

_t

t−1u(τ) dτ.

a Show that the system is real.

b Show that the system is stable.

c Show that the system is linear time-invariant.

d Calculate the response to the input u(t) = cos ωt.

e Calculate the response to the input u(t) = sin ωt.

f Calculate the amplitude response of the system.

g Calculate the frequency response of the system.

For a discrete-time system the response y[n] to an input u[n] is given by 1.10

y[n]= u[n − 1] − 2u[n] + u[n + 1].

a Show that the system is linear time-invariant.

b Is the system causal? Justify your answer.

c Is the system stable? Justify your answer.

d Calculate the frequency response of the system.

Two linear time-invariant continuous-time systems L₁and L₂are given with, re-1.11

spectively, frequency response H₁(ω) and H2(ω), amplitude response A1(ω) and A₂(ω) and phase response 1(ω) and 2(ω). The system L is a cascade connec-Cascade system

tion of L₁and L₂as drawn below.

u

L₁

y L₂

FIGURE 1.8

Cascade connection of L₁and L₂.

a Determine the frequency response of L.

b Determine the amplitude response of L.

c Determine the phase response of L.

For a linear time-invariant discrete-time system the frequency response is given by 1.12

H(e^iω) = (1 + i)e^−2iω.

a Determine the amplitude response of the system.

b Determine the response to the input u[n]= 1 for all n.

c Determine the response to the input u[n]= cos ωn.

d Determine the response to the input u[n]= cos²2ωn.

S U M M A R Y

An important field for the applications of the Fourier and Laplace transforms is signal and systems theory. In this chapter we therefore introduced a number of important concepts relating to signals and systems.

Mathematically speaking, a system can be interpreted as a mapping which assigns in a unique way an output y to an input u. What matters here is the relation between input and output, not the physical realization of the system.

Mathematically, a signal is a function defined onR or Z. The function values are allowed to be complex numbers.

In practice, various types of signal occur. Hence, the signals in this book were subdivided into continuous-time signals, which are defined onR, and discrete-time signals, which are defined onZ. An important class of signals is the periodic sig-nals. Another subdivision is obtained by differentiating between energy- and power-signals. Signals occurring in practice are mostly real-valued. These are called real signals. An important real signal is the sinusoidal signal which, for a given fre-quencyω, initial phase φ0and amplitude A, can be written as f(t) = A cos(ωt+φ0) in the continuous-time case and as f [n]= A cos(ωn +φ0) in the discrete-time case.

The sinusoidal signals are periodic in the continuous-time case. In general this is not true in the discrete-time case. A sinusoidal signal can be considered as the real part of a complex signal, the so-called time-harmonic signal ce^iωt or ce^iωn, with frequencyω and complex constant c.

Time-harmonic signals play an important role, on the one hand in all of the Fourier transforms, and on the other hand in systems that are both linear and time-invariant. These are precisely the systems suitable for an analysis using Fourier and Laplace transforms, because these linear time-invariant systems have the property that time-harmonic input result in outputs which are again time-harmonic with the same frequency. For a linear invariant system, the relation between a time-harmonic input u and the response y can be expressed using the so-called frequency response H(ω) or H(e^iω) of the system:

e^iωt→ H(ω)e^iωt (continuous-time system), e^iωn→ H(e^iω)e^iωn (discrete-time system).

The modulus of the frequency response, H(ω) or H(e^iω) respectively, is called the amplitude response, while the argument of the frequency response is called the phase response of the system. Of practical importance are furthermore the real, the stable and the causal systems.

Real systems have the property that the response to a real input is again real. The response of a sinusoidal signal is then a sinusoidal signal as well, with the same frequency.

Stable systems have the property that bounded inputs result in outputs that are also bounded. For these systems the frequency response is well-defined for eachω.

The response of a causal system at a specific time t depends only on the input at earlier times, hence only on the ‘past’ of the input. For linear time-invariant systems causality means that the response to a causal input is causal too. Here a signal is called causal if it is switched on at time t₀≥ 0.

S E L F T E S T

a Calculate the power of the signal f(t) = A cos ωt + B cos(ωt + φ0).

1.13

b Calculate the energy-content of the signal f(t) given by

f(t) =





0 for t< 0, sin(πt) for 0 ≤ t < 1,

0 for t≥ 1.

Show that the power of the time-harmonic signal f(t) = ce^iωtequals| c |². 1.14

a Calculate the power of the signal f [n]= A cos(πn/4) + B sin(πn/2).

1.15

b Calculate the energy-content of the signal f [n] given by f [n]=

 0 for n< 0, 1

2 n

for n≥ 0.

For a linear time-invariant continuous-time system the frequency response is given 1.16

by

H(ω) = e^iω ω²+ 1.

a Calculate the amplitude and phase response of the system.

b The time-harmonic signal u(t) = ie^{i t} is applied to the system. Calculate the response y(t) to u(t).

For a real linear time-invariant discrete-time system the amplitude response A(e^iω) 1.17

and phase response(e^iω) are given by A(e^iω) = 1/(1 + ω²) and (e^iω) = ω respectively. To the system the sinusoidal signal u[n]= sin 2n is applied.

a Is the signal u[n] periodic? Justify your answer.

b Show that the output is also a sinusoidal signal and determine the amplitude and initial phase of this signal.

For a continuous-time system the relation between the input u(t) and the corre-1.18

sponding output y(t) is given by y(t) = u(t − t0) +

_t

t−1u(τ) dτ.

a For which values of t₀is the system causal?

b Show that the system is stable.

c Is the system real? Justify your answer.

d Calculate the response to the sinusoidal signal u(t) = sin πt.

For a discrete-time system the relation between the input u[n] and the corresponding 1.19

output y[n] is given by y[n]= u[n − n0]+

n l=n−2

u[l].

a For which values of n₀∈ Z is the system causal?

b Show that the system is stable.

c Is the system real? Justify your answer.

d Calculate the response to the input u[n]= cos πn.

Mathematical prerequisites Introduction 27

2.1 Complex numbers, polynomials and rational functions 28 2.1.1 Elementary properties of complex numbers 28

2.1.2 Zeros of polynomials 32 2.2 Partial fraction expansions 35 2.3 Complex-valued functions 39 2.4 Sequences and series 45 2.4.1 Basic properties 45

2.4.2 Absolute convergence and convergence tests 47 2.4.3 Series of functions 49

2.5 Power series 51 Summary 55 Selftest 55