Input-Invariant Transformation

6.2 Input-Invariant Transformation

We will consider the input-invariant transformations that are error-free for specific input signals. For example, the impulse-invariant/step-invariant transformation is accurate when the input is an impulse or a step function. If they were accurate only for the specified inputs, they would be of limited practical value. However, by superposition, an input-invariant transformation gives zero error in response to any linear combination of specified input functions.

6.2 Input-Invariant Transformation 281

6.2.1 Impulse-Invariant Transformation

As mentioned in Remark 6.2, this is identical to the time-sampling method discussed in the previous section. Note that Eq. (6.1.8) can be written as

[n] ≡ g A (t ) | t =nT = g A (t ) ∗ δ(t)| t =nT (6.2.1) This implies that the (impulse) response of the continuous-time system G A (s) to the

impulse input δ(t ) is equal to the response of the discrete-time system G D [z] to the input (1/ T )δ[n], which is the sampled version of δ(t ). The procedure to derive the impulse-invariant equivalent G i mp [z] for a given analog system G A (s) is as follows:

1. Expand G A (s) into the partial fraction form:

2. Replace each term 1/(s − s i

Remark 6.4 Mapping of Stability Region by Impulse-Invariant Transformation Comparing Eq. (6.2.2) with Eq. (6.2.3), we observe that a pole at s = s i in s i T

the s-plane is mapped to z = e in the z-plane. Consequently, if and only if s i is in the left half plane (LHP), which is the stable region in the s-plane, then the corresponding pole is inside the unit circle, which is the stable region in the z-plane (see Fig. 6.2). However, the zeros will not in general be mapped in the same way as the poles are mapped.

10 x

s – plane

Im{z }

z=e sT

x 5 1 z – plane

xx

x Re{z } –10

Fig. 6.2 Mapping of poles from the s-plane to the z-plane

282 6 Continuous-Time Systems and Discrete-Time Systems Remark 6.5 Frequency Transformation by Impulse-Invariant Transformation

The relationship between the analog frequency ω A and the corresponding digital

frequency Ω is linear, that is, Ω = ω A T since

e jΩ

sT

s= jω A e jω A T (6.2.4) evaluation along the unit circle ≡ z=e evaluation along the j ω ≡ A −axis

Consequently, the shape of the frequency response is preserved. The negative aspect of this linear frequency relationship is that short sampling period T does not remove the frequency-aliasing problem caused by the impulse-invariant transformation (see Fig. 6.8).

Example 6.1 Impulse-Invariant Transformation – Time-Sampling Method For a continuous-time system with the system function G A (s) and frequency response G A ( j ω) as

jω+a the impulse-invariant transformation yields the following discrete system function

(E6.1.2) 1−e

6.2.2 Step-Invariant Transformation

then it will be an exact discrete-time equivalent of G A (s) for any input composed of step functions occurring at sample points. Note that Eq. (6.2.5) can be written as

(6.2.6) This implies that the step response of the continuous-time system G A (s) is equal to

the step response of the discrete-time system G st ep [z] on a sample-by-sample basis. That is why Eq. (6.2.5) is called the step-invariant transformation.

Let us consider the discrete-time error model for G A (s) and G st ep [z] in Fig. 6.3, in which the input to G A (s) is ¯x(t ), i.e., the zero-order-hold version of x(t ). The

6.2 Input-Invariant Transformation 283 y A (nT )

z.o.h.

G A (s)

x(t )

y A (nT ) – y step [n]

A /D

G step [z]

D/A

y step [n] Fig. 6.3 The discrete–time error model for the step–invariant transformation

discrete-time transfer function of the system with a z.o.h. in the upper part of Fig. 6.3 is

−T s

−1 {G h 0 (s)G A (s) }

≡G st ep [z] (6.2.7)

t =nT

Therefore the step-invariant transformation is also called the zero-order-hold equiv- alent mapping and it is well suited to a digital computer implementation in the A/D-G[z]-D/A structure.

Example 6.2 Step-Invariant Transformation (Zero-Order-Hold Equivalent) For a continuous-time system with the system function

G A (s) = with a pole at s = s p = −a (E6.2.1)

s+a

the step-invariant transformation yields the following discrete system function:

G (6.2.5) st ep [z] = (1 − z −1 ) Z

1−e −aT

−aT z−e =G This implies that the s-plane pole is mapped into the z-plane through the step-

G A (s)

zoh [z] (E6.2.2)

invariant transformation in the same way as through the impulse-invariant trans- formation.

s s=s T

= −a → z = z p =e =e −aT

284 6 Continuous-Time Systems and Discrete-Time Systems