Page 877 - The Mechatronics Handbook

0066-frame-C29 Page 7 Wednesday, January 9, 2002 7:23 PM

where Z[H(s)/s] represents the z-transform of the sampled version of the step response of the continuous-
time system. The form for a generic first-order system is given below:

−aT
−1
)z
a
1 e
–
Hs() = ----------- ⇔ H d z() = ( ------------------------------
s + a 1 e −aT −1
z
–
The response matching method (especially with step inputs) is commonly used to map a continuous-
time plant to discrete-time when designing a digital controller in the discrete-domain. Since most digital
controllers are implemented using a ZOH on the output of the digital controller, the plant sees a stepped
signal, one that looks like a sum of delayed step signals. Therefore, the step-response matching method
is the most accurate way to map a plant that has a ZOH on its input.
Frequency Domain Mappings

The continuous-time Fourier transform can be related to the DTFT through the expression:
X w() = TX Ω()| Ω=w T for −p ≤ Ω ≤ p

where X(ω) is defined in Eq. (29.1) and represents the continuous-time Fourier transform of x(t), while
X(Ω) is defined in Eq. (29.2) and represents the DTFT of the sampled signal x[n]. This mapping is very
useful for computing the Fourier transform of measured data. In particular, suppose a continuous-time
signal is measured by sampling it through an ADC and storing it as a discrete-time sequence. If the signal
x(t) is finite in duration, the DTFT of x[n] can be computed at discrete points in frequency by using the
, where
DFT, X k , as given in Eq. (29.4). Using the relationships ω = Ω/T, Ω = 2πk/N, and X k = X(Ω)| Ω=2pk/N
N is the length of the sequence for x[n] and T is the sampling period, gives the relationship:
X w()| w =2pk/NT = TX k for 0 ≤ w ≤ w s /2, 0 ≤≤ ( N 1)/2
–
k
where ω s = 2π/T is the sampling frequency in radian per second. Accuracy can be improved by decreasing
the sampling period T, and the resolution in the plot can be increased by increasing NT.
If the signal x(t) is not finite in duration, it must be truncated in order to use this numerical method
to calculate the continuous-time Fourier transform. As discussed in the section “Discrete-Time Signals,”
if the sampled signal x[n] decays to zero, choose the number of sampled points N to be large enough so
that x(t) is negligible beyond that value. If the sampled signal x[n] is periodic, choose the sampling period
T and the number of points N such that the sampled signal x[n] goes through an integer number of
cycles. For example, consider the signal x(t) = cos(πt). If the sampling period is chosen as T = 0.4 s, the
discretized signal would be x[n] = x(nT) = cos(0.4πn), which is the same signal analyzed in the section
“Discrete-Time Signals.” Choosing N = 5, 10, 15, etc. would yield correct results in the DFT while any
other value would result in leakage. If the signal x(t) has periodic content, but does not appear to be
periodic, then use a windowing function as discussed in the section “Discrete-Time Signals” to reduce
leakage when computing the DFT.
Note that the DFT can also be used to determine the Fourier coefficients of periodic signals. Consider
a Fourier series in the form
∞
xt() = ∑ c k e jw kt
k=−∞
Sample the signal x(t) by first making sure that the sampled signal goes through an integer number of
cycles. The coefficients c k for k = 0,…,(N − 1)/2 can be found as c k = X k /N where X k is found from the
∗
DFT. The rest of the coefficients are obtained from c −k = c k .

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