Linear discrete-time models Difference equations

Let f ltl denote the (scalar) input to a discrete-time linear system, and let y [ t ] denote the (scalar) output. It is common to assume an inputloutput relation of the form of the difference equation

y [ t l = - Z i y [ t

-

I] - - Z z y [ t -21 - . . , - a , y [ t

-

- p ] + F o f [ t ]

+ I f ; , f [ t - l ] + . . . + b , f [ t - q ] . (1.1) The equation is shown under general assumption of complex signals, and the bar over the coefficients denotes complex conjugation. (See box 1.1 ^.) By redefining each coefficient El and

Fl

in terms of its conjugate, (1.1) could also be written without the conjugates as

With consistent and careful use of the notation, the question of whether the coefficients are conjugated in the definition of the linear model is of no ultimate significance-the answers obtained are invariably the same. However, the bulk of signal processing literature seems to favor the conjugated representation in ( I . l ) , and we follow that convention. Of course,

Box 1.1: Notation for complex quantities

We use the engineer's notation j =

a,

rather than the mathematician's i . However, in some places j will be used as an index of summation; context should make clear what is intended.

A bar over a quantity denotes complex conjugation. Other authors com- monly indicate complex conjugation using a superscript asterisk, as a*. How- ever, the Z i notation is used in this book to indicate conjugation, since a* is also commonly used to denote a particular value of a , such as a minimizing value, or to indicate the adjoint of a linear operator.

8 Introduction and Foundations

when the signals and coefficients are strictly real, the conjugation is superfluous and the system can also be written in the form

~ [ t ] = -aly[t

-

11 - aay[t

-

21 -

. . . -

a,y[t

-

p]

+

^{bo f [t]}

+ b 1 f [ t - l l + . . . + b q f [ t - q ] without the conjugates on the coefficients.

In the case of a system that is not time invariant, the coefficients may be a function of the time index r. We will assume, for the most part, constant coefficients. The relation (1.1)

can be wriiten as P 4

x 4 y l t - k l = x F k f [ t k ] , with ~o = 1.

In (1.2), when p = 0,

the signal y[t] is called in the statistical literature a moving average (MA) signal, since it is formed by simply adding up (scaled versions of) the input signal over a window of q

+

1 values. The number q is the order of the MA signal. The signal is denoted either as MA or MA(q). We can also write (1.3) using a convenient vector notation. Let

Then

The vector notation used in this example is summarized in box 1.2. In equation 1.2, when q = 0, so that

P

~ [ t l =

%f

- x a k Y [ t

-

k],

k = l

the signal y is said to be an autoregressive (AR) signal of order p. Auto because it expresses the signal in terms of itself; regressive in the sense that a functional relationship exists between two or more variables. An autoregressive model is denoted as AR orAR(p). Writing

we can write the AR signal as

The general form in (1.2), combining both the autoregressive and the moving average components, is called an autoregressive moving average, or ARMA, orARMA(p, q). Where all the signals are deterministic, the term DARMA (deterministic ARMA) is sometimes employed.

1.4 'Clociels for Idinear Sv5tern5 and Signals 9

1

Box 1.2: Notation for vectors

1 Vector\ i n ;I hnrte-d1nlen\ion;t1 vector ipace ;Ire denoted In bold font, such ,ii b

2 All vectors in t h ~ \ book &re a\iumed to be column vectori In some c a m ^'1 vector wlll be typeiet In h o r ~ ~ o n t d l formdt, wrth

'

(tranipore) to indicate that i t should be trancpoced Thui we could h u e equivalently written

3 In general, the ~ t h component of a vector b will be des~gnated as b, Whether the tndex r itarts with 0 or I (or some other value) depends on the needs of the partrcular problem

4 The notat~on b" denote\ the Herrriltlan tran\po\e, in which b ts trans- poied and tts elements are conjugdted

-

bH =

[go, gi,

, bql

These rules notwithstanding, for notational conventence we will sometimes denote the vector with Y Z elements as an n-tuple, so that

I

x = [ x [ x? . . . .rn] and x = ( x , , x ~ , . . . , n,)

are occasionally used synonymously. This n-tuple notation is used particularly when x is regarded as a point in Rn. Furthermore, since we will generalize the concept of vectors to include functions, the math italic notation x will be used in the most general case to represent vectors, either in

Rn

or as functions.

Matrices are represented with capital letters, as in A or X. The matrix I is an identity matrix. The notation 0 is used to indicate a vector or matrix of zeros, with the size determined by context. Similarly, the notation 1 is used to indicate a vector or matrix of ones, with the size determined by context.

System function and impulse response

In the interest of getting a system function that does not depend upon initial conditions, we assume that the initial conditions are zero, and take the Z-transform to obtain

which we write as

Y(z)A(;) = F ( r ) B ( z ) We will occasionally write the transform relationship as

where the particular transform intended is determined by context. We will also denote 2-transforms by

10 Introduction and Foundations

The system function is

This is also called (usually interchangeably) the trar~sfer funcfion of the system. We write

Y ( z ) = H ( z ) F(z), (1.5)

and represent this as shown in figure 1.1. If the system is AR, then

Figure 1.1 : Input /output relation for a transfer function

and H (z) is said to be an all-pole system. If the system is MA, then

which is called an all-zero system. The corresponding difference equation (1.3) has only a finite number of nonzero outputs when the input is a delta function f [ t ] = S [ t ] , where

We will also write the delta function as 8,. Occasionally the function S [ r

-

^{t ]} will be wntten as

&,,.

A system that has only a finite number of nonzero outputs in response to a delta function is referred to as a finite impulse response (FIR) system. A system which is not FIR is infinite impulse response (IIR).

We can view signal Y(z) as the output of a system with system function H(z) driven by an input F(z). Taking the inverse Z-transform of ( 1 . 3 , and recalling the convolution property (multiplication in the transform domain corresponds to convolution in the time domain) we obtain

? ; [ I ] = f [ k ] h [ r - k ] .

where h [ r ] , the impulse response, is the inverse transform of H(z).

To compute the inverse transform of H(z). we first factor H(z) into monomial factors using the roots of the numerator and denominator polynomials.

where the

z,

are the nonzero roots of B(z) (called the :el-05 of the system function) and the p, are the nonzero roots of A ( ? ) (called the poles of the system function) In this form, we observe that fa pole IS equal to a rtero, the factors can be canceled out of both the numerator and denominator to obtain an equivalent transfer functlon A word of caution even though terms may cancel from the numerator and denominator as seen from the trantfcr tunct~on, the physical components that the5e term5 model Indy 51111 exist and could introduce difficulty

1 ,J Models for Linear Systems and Signals 11

A system with the smallest degree numerator and denominator is said to be a ^minimal system.

Example 1.4.1 The system function

can be factored as

Thus the H ( z ) is not a minimal realization.

Partial Fraction Expansion (PFE)

Assuming for the moment that the poles are unique (no repeated poles) and that q < p , then, by partial fraction expansion (PFE), the system function can be expressed as

where

Taking the causal inverse Z-transform of (1.6), we obtain

The functions pi are the natural modes of the system

N (z).

Clearly, for the causal modes to be bounded in time, we must have Ipk 5 1. In general, the output of a linear time-invariant system is the sum of the natural modes of the system plus the input modes of the system.

Example 1.4.2 Let

Then, a partial fraction expansion is

The impulse response is

hit1 = [(-2)(.5)'

+

3(.6)']u[t], where u [ t ] is the unit-step function,

To compute the PFE when q 2 p , the ratio of polynomials is first divided out. When there are repeated poles, somewhat more care is required. For example, a root repeated r

12 Introduction and Foundations

times, as in

gives rise to the partial fraction expansion

where'

The inverse Z-transform corresponding to (1.7) is of the form

where the coefficients {c,) are linearly related to the PFE coefficients {k,).

Using computer software, such as the residue or res iduez command in MATLAB, is recommended to compute partial fraction expansions.

Example 1.4.3 Let

We desire to find the impulse response h [ t ] . Since the degree of the numerator is the same as the degree of the denominator. we divide, then find the partial fraction expansion.

then,

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