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1 Do the following for a fifth-order Butterworth filter with cut-off frequency of 1 kHz and
transfer function B(s).
(a) Write the expression for the magnitude squared of the frequency response.
(b) Sketch the locations of the poles of B(s)B(-s).
(c) Indicate the locations of the poles of B(s), assuming that B(s) represents a causal and
stable filter.
(d) Indicate the locations of the poles of B(-s).
2 Figure P24.2-1 shows the frequency response of a discrete-time filter.
(a) Determine and sketch the analog frequency response characteristic that (assuming no
aliasing) will map to the discrete-time frequency response given in the figure when the
impulse invariance method is used.
PROBLEMS
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(b) Sketch the analog frequency response that will map to the discrete-time frequency
response in Figure P24.2-1 when the bilinear transformation is applied.
(c) Repeat parts (a) and (b) for the discrete-time frequency response characteristic in Figure
P24.2-2.
3 Consider the system function
(a) Determine the discrete-time transfer function Hd(z) obtained by mapping Hc(s) to
Hd(z) using the bilinear transformation with T = 2.
(b) Find the range of the constant a for which H(s) is stable and causal.
(c) Verify that if H(s) is stable and causal, then H(z) is also stable and causal.
4 Consider the following discrete-time filter specifications:

To design Hd(eju) using the impulse invariance method or the bilinear transfor mation, we
need first to specify a continuous-time filter B(jw). Assume that we will use a Butterworth
filter.
(a) Set up the proper equations for the order N and the cutoff frequency we of the
continuous-time filter B(jw) that will map to Hd(e'n) when the impulse invari ance
method is used. Set T = 1.
(b) Set up the proper equations for N and w, of B(jo) when the bilinear transfor mation is
used. Set T = 1.
5 Consider the system in Figure P24.5, which implements a continuous-time filter by
discrete-time processing.
The sampling frequency is 15 kHz. The continuous-time filter must satisfy the following
specifications:

(a) Determine the appropriate specifications for H(ej"), the frequency response of the
discrete-time filter.
(b) Suppose that to design H(ej"), we use the impulse invariance method. We need to
introduce a second continuous-time filter, G(jw). Using T = 3 for the value of the
parameter T in the impulse invariance design procedure, determine the filter
specifications of G(jw).
(c) Suppose that we now use the bilinear transformation to design H(ej). Using T = 2 for
the value of the parameter T in the bilinear transformation method, determine the
filter specification of G(jw).
(d) With either the impulse invariance or bilinear design procedure, is H(eju) or H,(jw)
dependent in any way on the parameter T?
6 In this problem we consider more closely the design procedure for continuous-time
Butterworth filters.
Suppose that we are to design a filter B(jw) such that

(a) There are two unknown parameters, the order N of B(s) and the cutoff fre quency co,.
Set up the two simultaneous equations for N and co, and verify that N = 5.88 and w, =
7.047 satisfy the equations.
(b) Since N is not an integer, we must choose N to be the next higher integer. We can now
pick whether to meet exactly the stopband specification and exceed the passband
specification or vice versa. Find w, such that the passband specifica tion is met
exactly, and verify that the stopband specification is exceeded.
(c) What would happen if we picked N = 5?
7 We want to design a discrete-time lowpass filter with a passband magnitude char
acteristic that is constant to within 0.75 dB for frequencies below 0 = 0. 2613x and
that has a stopband attenuation of at least 20 dB for frequencies between Q = 0.4018x
and r. Determine the poles of the lowest-order Butterworth continuoustime transfer
function that, when mapped to a discrete-time filter using the bilinear transformation
with T = 1, will meet the specifications. If possible, exceed the stopband
specifications. Indicate also how you would proceed to obtain the transfer function of
the discrete-time filter.
8 Suppose that we want to design a discrete-time filter using the impulse invariance
method. The filter specifications are given by

Using T = 3, we obtain the corresponding filter specifications for the associated
continuous-time filter Hb(s):
Assume that a filter H,(s) satisfies the specifications exactly; thus,
The designed discrete-time filter is given by
Study the case of T = 2.
(a) For T = 2, give the filter specifications for the associated continuous-time filter I(s).
(b) Verify that the continuous-time filter given by

satisfies part (a) exactly.
(c) Substitute 11,(s) from part (b) to solve for H(eja) and verify that H(ei") = H(eju). Thus,
the value of T does not affect the final discrete-time filter designed.
9. As mentioned in Section 10.8.3 of the text, the bilinear transformation map from the s
plane to the z plane can be interpreted as arising from the use of the trapezoidalrule in
numerically integrating differential equations.
(a) Consider a continuous-time system for which the differential equation is
or, equivalently,
Determine the system function H(s) for this continuous-time system.
In numerical analysis the procedure known as the trapezoidal rule for integration
proceeds by approximating the continuous-time function as a set of contiguous
trapezoids, as illustrated in Figure P24.9(a), and then adding their areas to compute the
total integral. The areaA of an individual trapezoid, with dimensions shown in Figure
P24.9(b), is

(b) What is the area An in the trapezoidal approximation between x[(n -1)T] and x(nT)?
(c) From eq. (P24.9-2), y(nT) denotes the area under x(t) up to time t = nT. Let y[n] denote
the approximation to y(nT) obtained using the trapezoidal rule for integration, that is,

Show that
(d) With -V[n] defined as &[n] = x(nT), show that the trapezoidal rule approxima tion to
eq. (P24.9-2) becomes
(e) Determine the system function corresponding to the difference equation in part (d).
Demonstrate, in particular, that it is the same as would be obtained by applying the
bilinear transformation to the continuous-time system function corresponding to eq.
(P24.9-1).

SOLUTIONS
S.1 (a) For N = 5 and we = (2π)1 kHz, IB(jw)І2 is given by
(b) The denominator of B(s)B(-s) is set to zero. Thus
as shown in Figure S24.1-1.

(c) For B(s) to be stable and causal, its poles must be in the left half-plane, as shown in
Figure S24.1-2.
(d) Since the total number of poles must be as shown in part (b), the poles B(-s) must be
given as in Figure S24.1-3.

S.2 (a) When there is no aliasing, the relation in the frequency domain between the
continuous-time filter and the discrete-time filter corresponding to impulse invariance is
Thus, there is an amplitude scaling of T and a frequency scaling given by
The required transfer function can be found by reflecting TH(e'u) through the preceding
transformation, as shown in Figure S24.2-1.

Since the relation between 0 and wis linear, the shape of the frequency response is
preserved.
(b) For the bilinear transformation, there is no amplitude scaling of the frequency response;
however, there is the following frequency transformation:
As in part (a), we can find H(jw) by reflecting H(ej") through the preceding frequency
transformation, shown in Figure S24.2-2.

Because of the nonlinear relation between Q and w, H,(jo) does not exhibit a linear slope
as H(e'") does.
(c) We redraw the transformation of part (a) for the new H(e'0 ) in Figure S24.2-3. As in
part (a), the shape of the frequency response is preserved.
We redraw the transformation of part (b) for the new H(ej") in Figure S24.2-4. Unlike part
(b), the general shape of H(eju) is preserved because of the piece wise-constant nature of
H(ej").

S.3 (a) Using the bilinear transformation, we get
(b) Since H(s) has a pole at -a, we need a > 0 for H(s) to be stable and causal.
(c) Figure S24.3 contains a plot of (1 -a)/(1 + a), the pole location of H(z), ver sus a.

We see that for a > 0, (1 -a)/(1 + a) is between --1 and 1. Since the only pole of H(z)
occurs at z = (1 -a)/(1 + a), H(z) must be stable whenever H(s) is stable, assuming that
H(z) represents a causal h[n].
S.4 (a) For T = 1 and the impulse invariance method, B(jw) must satisfy

Therefore, if we ignore aliasing,
(b) For T = 1 and the bilinear transformation, B(jw) must satisfy
Therefore,

S.5 (a) The relation between 0 and wis given by Q = wT, where T = 1/15000. Thus,
Note that while Hd(jw) was restricted to be between 0.1 and 0 for all o larger than
2r(4500), we can specify H(eja) only up to Q = x. For values higher than w, we rely on
some anti-aliasing filter to do the attenuation for us.
(b) Assuming no aliasing,
Therefore,

(c) The relation between o and Qis given by Q = 2 arctan (w). Thus,
(d) If T changes, then the specifications for G(jw) will change for either the impulse
variance method or the bilinear transformation. However, they will change in such a way
that the resulting discrete-time filter H(e'") will not change. Thus,He(jo) will also not
change.
S.6 (a) We first assume that a B(s) exists such that the filter specifications are met
exactly. Since
we require that

Substituting N = 5.88 and co, = 7.047, we see that the preceding equations are satisfied.
(b) Since we know that N = 6, we use the first equation to solve for we:
Solving for co,, we find that w, = 7.032. The frequency response at ( = 0.31 is given by
(c) If we picked N = 5, there would be no value of we that would lead to a Butterworth filter
that would meet the filter specifications.
S.7 We require an Hd(z) such that

We will for the moment assume that the specifications can be met exactly. Let 9, be the
frequency where
Similarly, we define Q, as the frequency where
Using T = 1, we find the specifications for the continuous-time filter Ha(jo) as
Where
For the specification to be met exactly, we need N and co such that
Solving for N, we find that N = 6.04. Since N is so close to 6 we may relax the specifications
slightly and choose N = 6. Alternatively, we pick N = 7. Meeting the passband specification
exactly, we choose w,such that

The continuous-time filter H,(s) is then specified by
The poles are drawn in Figure S24.7.

We associate with Ha(s) the poles that are on the left half-plane, as follows:
Hd(z) can be obtained by the substitution
S.8 (a) Assuming no aliasing, Hd(e'u) is related to Hb(jw) by

(b) Substituting

S.9 (a) Using properties of the Laplace transform, we have
(b) Here h is given by T, a is given by x[(n -1)T], and b is given by x(nT). There fore, the
area is given by
(c) From the definition of 9[n], we find that

(d) From the answer to part (a), we substitute for An, yielding
(e) Using z-transforms, we find

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