iified

put of ,led at at the

)Ur the times. in this : it can

~ partly , y~tems

rs. The

a linear

. sample

equency

Istructed

'y block-

Ipossible

be com­7s/orm, the must

Figure 5.19 Impulse response of a first-order hold filter

Figure 5.20 Block diagram of a sampleand first-order hold

Figure 5.21 Asampled-data system

5.7 Problems 183

S.2 Show how to construct a signal of hidden oscillations that grows in an unstable fashion . Where in the s-plane are the poles of the transforms of your signal(s)?

S.3 A first-order hold is a device that extrapolates a line over the interval from k T to (k + 1) T with slope given by the samples r(kT ­ T) and r(kT) starting from r(kT) as shown in Fig. 5.19. Compute the frequency response of the first-order hold.

S.4 Consider the circuit of Fig. 5.20. By plotting the response to a signal that is zero for all sample instants except I = 0 and that is \.0 at I = 0, show that this circuit implements a first-order hold .

S.S Sketch the step response y(t) of the system shown in Fig. 5.21 for K = ~, I , and 2.

S.6 Sketch the response of a second-order hold circuit to a step unit. What might be the major disadvantage of this data extrapolator?

S.7 A triangle hold is a device that has an output, as sketched in Fig. 5.22 that connects the samples of an input with straight lines.

3

2

0

- I

-2 0

(a) Sketch the impulse response of the triangle hold. Notice that it is noncausal.

(b) Compute the transfer function of the hold.

(c) Use MATLAB to plot the frequency response of the triangle hold.

I I I I I I -----1.----+---­

I I I I 1 I

----~---J----~----I I I I I I

===tsJ~~~l~~~ I

I I I I

2 3 4

~

184 Chapter 5 Sampled-Data Systems

Figure 5.22 Response of a sample and triangle hold

(d) How would the frequency response be changed if the triangle hold is made to be cau sal by adding a deJay of one sample period?

5.8 Sketch the output of a sample and zero-order hold to

(a) A step input.

(b) A ramp input.

(e) A sinu soid of f requency w,l IO .

5.9 Sketch the output of a sample and first-order hoJd to

(a) A step input.

(b) A ramp input.

(e) A sinusoid of frequency ws/ IO.

5.10 Sketch the output of a sa mple and triangle hold to

(a) A step input.

(b) A ramp inpu t.

(e) A sinusoid of frequency w,l lO.

5.11 Sketch the output of a sample and causal triangle hold to

(a) A step input.

(b) A ramp input.

(e) A si nusoid of frequency w) I O.

5.12 A sinusoid of frequency 11 rad/sec. is sampled at the frequency Ws = 5 rad / sec.

(a) Indicate the component frequencies up to w = 20 rad/ sec.

(b) Indicate the relative amplitudes of the components up to 20 rad / sec. if the sampler is followed by a zero-hold.

5.13 A signal r(t) = sin(2!) + sin (J5t) is sampled at the frequency Ws = 16.

(a) Indicate the frequency of the components in the sampled signal up to (I) = 32.

(b) Indicate the relative amplitudes of the s ignals in the output if the s ignal is passed through the anti-aliasing fi lter wi th transfer function _,_1_, before samplinb(J. You

(~ " 1) ­

can use MATLAB Lo compute the filter gain. '

5.14 Derive Eq. (5 .38).

5.15 Find the transform of the output Y (s) and its samples Y' (5) for the block diagrams shown in Fig. 5.23. Indicate whether a transfer fu nction ex ists in each case.

5.16 Assume the followin g transfer func tions are preceded by a sampler and zero-order hold and followed by a sampler. Compute the res ulting discre te transfer functions.

Fig BIoI san­(a)

(b) I

Figu

B the

Figure 5.23 Block diagra ms of sampled data systems. (a) Single loop; (0) mul tiple loop

lef

~d

)u

Figure 5.24 ,wn Block diagrams showing

:he modified z-transform

old

5. 7 Problems 185

~ M _y . ---o~'-- _ / y * -f 'ci};IT (a)

y* ---0

(b)

(a ) G (s ) = l /s2 1

(b) G (s) = e- 151 / (s + 1)2

(e) G,(s) = l / s (s +l )

(d ) G (s) = e- 1 5, / s(s + l ) 4

(e) Gs(s) = 1/ (s2 - I)

5.17 One technique for examining the response of a sampled data sys tem between sampling in stants is to shift the response a f rac tion of a period to the lef t and to sa mple [he result. The e ffec t is as shown in the block diagram of Fig , 5.24 and described by the equation

Y* (s. m) = R" (s )(G(s )emTsj*.

As a func tion of z, the equiva lent equation is

Y(z ) = R(z) G (z , m).

The fun ction G(z, m) is ca lled the modified z - t ransform of G(s). In the figure, let

1 1 G (s) = - - , R (s) = - , and T = 1.

(5 + 1) s

(a) Compute y( l ) by calculating y(kT) from the ordinary z - t ransf orm anel o bserving tha t between samples , the output y (t ) is an exponential decay with

R /~*TY Y* (s,m)~ G(s) em s ---0

T T y (t) y (t + mD

O::;;;m < 1

6.5 Problems 209

6.5 Problems

dis- 6.1 Sketch the zone in the z-plane where poles corresponding to the left half of the s-plane ods will be mapped by the zero-pole mapping technique and the zero-order-hold technique.

ms. 6.2 Show that Eq. (6. 15) is true.

6.3 The following transfer function is a lead network designed to add about 60° phase lead at WI = 3 rad

s + J H (s) = - --.

O.Is + I (a) For each of the following design methods compute and plot in the z-plane the pole

and zero locations and compute the amount of phase lead give n by the equivalent network at ZI = e jw1T if T = 0.25 sec and the design is via

l. Forward rectangular ru Ie

ii. Backward rectangular rule

iii. Bilinear rule

IV. Bilinear with prewarping (use WI

v. Zero-pole mapping

vi. Zero-order-hold equivalent

vii . Triangular-hold equivalent

as the warping frequency )

(b) Plot over the frequency range WI = 0.1 ~ lJ)h = 100 the amplitude and phase Bode plots for each of the above equivalents.

6.4 The following transfer function is a lag network designed to increase Kv by a factor of 10 and have negligible phase lag at W I = 3 rad.

·ete lOs + 1ing H (s) = J 0 100s + I .

the (a) For each of the following design methods, compute and plot on the z-p lane thelod

zero-pole patterns of the resulting discrete equivalents and give the phase lag at wn

Zl = ejU)1 T corresponding to WI = 3. Let T = 0.25 sec. md

i. Forward rectangular rule "all

ii. Backward rectangular ruletly iii. Bilinear rule eet

on iv. Bilinear with prewarping (Use IJ)I = 3 radians as the warping frequency)

)m v. Zero-pole matching

vi. Zero-order-hoJd equ iva lent

vii. Triangle-hold equivalent

(b) For each case computed, plot the Bode amplitude and phase curves over the range WI = 0.01 ~ wI, = 10 rad.