Concept of Discrete-Time Equivalent

6.1 Concept of Discrete-Time Equivalent

As the technology of digital processors becomes more advanced, it is often desirable to replace analog systems with digital ones. In some cases, we need to transform the analog systems into their “equivalent” digital systems in order to simulate their behaviors on digital computers. In other cases, rather than designing directly a

278 6 Continuous-Time Systems and Discrete-Time Systems digital system, we make use of a variety of well-developed analog system design

procedures to get an analog system with satisfactory performance and then convert it into a discrete-time equivalent that mimicks (emulates) the analog system. In

either case, for a given continuous-time linear time-invariant (LTI) system G A (s) with input x(t ) and output y A (t ), we wish to find a discrete-time system G D [z] with input x[n] and output y D [n] such that

x [n] = x(nT ) ⇒ y D [n] = y A (nT ) (6.1.1) This means that G A (s) and G D [z] yield the equivalent outputs to equivalent inputs

in the discrete-time domain. We will call such a system G D [z] the discrete-time (z-domain) equivalent or digital simulator for G A (s) where Eq. (6.1.1) is referred to as the (discrete-time) equivalence criterion. To establish necessary conditions for the validity of the above criterion, we shall first consider sinusoidal inputs, which will lead to the general case, because an arbitrary signal can be expressed as a linear combination of sine waves (the Fourier series representation). Suppose the input

x [n] of a discrete-time system G D [z] is the discrete-time version of x(t ) = e j ωt sampled with period T , i.e., x[n] = x(nT ) = e j nωT where x(t ) is the input to the

continuous-time system G A (s) (see Fig. 6.1):

(6.1.2) Then we can use the convolutional input-output relationships (1.2.4)/(1.2.9) of

(t ) = e j nωT −−−−−−−−−−−−−→ x[n] = x(nT ) = e

j ωt

Sampling with period T

continuous-time/discrete-time LTI systems and the definitions of CTFT and DTFT

to write the outputs of G A (s) and G D [z] as

A (t ) =

g A (τ )e (t−τ )

jω

dτ

[m]e D jω ? (n−m)T y D [n] = g

−−−−−−→ m=−∞

A ( j ω)e j ωt =G D [e ]e (6.1.3) where

(2.2.1a)

=G j nωT

Sampling with period T

(3.1.1) j ωT

G (A.1)

causal system (τ )=0 ∀ τ <0

g D m=−∞ D

G [z]

causal system [n]=0 ∀ n<0

Fig. 6.1 The equivalence criterion between a continuous-time system and a discrete-time system

6.1 Concept of Discrete-Time Equivalent 279 The above equivalence criterion (6.1.1) can be stated in terms of the frequency

responses of the continuous-time and discrete-time systems as

G D [e j ωT ]=G A ( j ω)

This is referred to as the frequency-domain discrete-time equivalence criterion. However, this equivalence criterion cannot be satisfied perfectly because the digital frequency response G D [e j ωT ] is periodic in ω (Sect. 3.2.1) while the fre- quency response G A ( j ω) is not. This implies that we cannot find the discrete-time equivalent of a continuous-time system that would work for all kinds of inputs. In principle, the criterion (6.1.5) can be satisfied only for the principal frequency range (−π/T, +π/T ) if we restrict the class of inputs into band-limited signals that do not contain frequencies higher than the folding frequency π/ T , i.e., half the sampling

frequency ω s = 2π/T . In this sense, the resulting system G D [z] satisfying the criterion (6.1.5) is a discrete-time equivalent of G A (s), which is valid only for such band-limited input signals.

Remark 6.1 Equivalence Criterion and Band-Limitedness Condition

(1) The band-limitedness condition is not so restrictive since we can increase π/ T by decreasing the sampling period T . Furthermore, the condition is desirable in order for the sampled output y(nT ) to represent y(t ) faithfully.

(2) As a matter of fact, the equivalence criterion (6.1.5) cannot be met exactly even for the principal frequency range since G D [e j ωT ] is a rational function of

e j ωT , while G A ( j ω) is a rational function of j ω. For engineering applications, however, approximate equality is good enough.

Now we will discuss the time-sampling method, which determines G D [z] so as to satisfy the criterion (6.1.5) with reasonable accuracy for almost every ω in the frequency band of the input signal. The basic idea is as follows. Suppose that the

input x(t ) and the impulse response g A (t ) of the analog (or continuous-time) system

G A (s) are sufficiently smooth, i.e., nearly constant in any interval of length T . Then, letting

g D [n] = T g A (nT )

satisfies approximately the equivalence criterion (6.1.1), i.e.,

y D [n] ∼ =y A (nT ) for x[n] = x(nT ) since we have

280 6 Continuous-Time Systems and Discrete-Time Systems

y A (nT ) =

g A (τ )x(t − τ )dτ

(nT )x(nT − mT ) (6.1.7a)

[m]x[n − m] (6.1.7b)

m=−∞

The condition (6.1.6) can be written in terms of the transforms as

G D [z] = T Z{g A (nT )} = T Z

−1 {G A (s)}| t =nT (6.1.8)

Remark 6.2 Time-Sampling Method – Impulse-Invariant Transformation Note that sampling the continuous-time unit impulse δ(t ) ∼ = (1/T )sinc(t/T ) (Eq. (1.1.33a)) at t = nT (i.e., every T s) yields (1/T )sinc(n) (1.1.35) = (1/T )δ[n].

Then, it is very natural that, in order for G A and G D to be similar, their responses to δ (t ) and (1/ T )δ[n] should be the same, i.e.,

g A (nT ) = G A {δ(t)}| t =nT ≡G D δ [n] = G D

{δ[n]} = g D [n] T (6.1.9)

where G{x(t)} denotes the output of a system G to an input x(t). This is equivalent to the condition (6.1.6) for the impulse-invariant transformation.

Remark 6.3 Frequency Response Aspect of Impulse-Invariant Transformation If G A ( j ω) is negligible for |ω| > π/T (in agreement with the smoothness conditions about g A (t )), then we can take the DTFT of (6.1.6) and use Eq. (3.5.7) or (5.3.1b) to get

(5.3.1b)

G D (Ω) = F{g D [n]}| Ω =ωT ≡ T F(g A (nT )} = ∼ G A (ω) (6.1.10)