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Problems
Problem 13.1
H(z) represents the system function for an FIR linear system, and is given by
Draw a flow-graph implementation of the system in each of the follow ing forms:
(i) cascade for
(ii) direct form
(iii) linear-phase form
(iv) frequency-sampling form
Problem 13.2
We wish to implement a digital oscillator, i.e., a digital filter for which the
unit-sample response is of the form
Shown in Figure P13.2-1 are a direct form and a coupled form second order
filter.

(a) Assuming that there is no parameter quantization, determine the coefficients a and b so that the unit-
sample response of the direct form filter will be of the form of Eq. (13.2-1) with
(b) Again, assuming that there is no parameter quantization determine the coefficients a2 and b2 so that
the unit-sample response of the coupled form filter will be of the form of Eq. (13.2-1) with w0 Although
the amplitude A and phase shift $ will be different for the direct form and coupled form, this difference
is not important for this problem.
(c) Now, let us quantize the coefficients for the two filters. In particular, the fractional part of the absolute
value of each co efficient will be represented by, at most three bits. The value of the quantized
coefficient can be obtained as follows:
(i) multiply the coefficient by 8
(ii) discard the fractional part of the product
(iii) divide the result in (ii) by 8.
Determine whether, after quantization the unit-sample response of each filter will still be of the form of
Eq. (13.2-1). If yes, determine the new value of 0'

Problem 13.3
A causal linear-phase FIR system has the property that h(n) = h(N - 1 - n) for n = 0,1..., N - 1.
This symmetry constraint was used in Sec. 6.5.3 of the text to show that systems satisfying
this constraint have linear phase corresponding to a delay of (N - 1)/2 samples. This constraint
results in a significant simplication of the frequency-sampling realization of Eqs. (4.48) and
(4.50) of the text.
(a) Using the above linear-phase constraint, show that, for N even, H(N/2) = 0. (b) Consider
H(k) expressed in the form H(k) = H (k)ejO(k) where a Ha(k) is'real. Determine an
expression for 0(k) for k = 0, 1,..., N - 1 that is valid for N even. You may find it helpful to
refer to the results in Sec. 6.5.3 of the text.
(b) (c) Assume that Ha(k) defined in (b) is position i.e. that it corresponds to the magnitude of
H(k). Using the results of part (a) and (b), show that for h(n) linear phase and with N even,
Eqs. (4.49) and (4.50) of text can be simplified to
(We have assumed for convenience that r = 1.)
(d) Draw a flow-graph representation of the system function derived
in part (c).

NETWORK STRUCTURES FOR FIR SYSTEMS AND PARAMETER QUANTIZATION
EFFECTS IN DIGITAL FILTER STRUCTURES
Solution 13.1
(i) Since H(z) has only real zeros we will use only first-order sections in the cascade form.
Then Figure S13.1-1 represents one possible ordering for these sections.
Figure S13.l-1
(ii) For the direct form we first express H(z) as
The direct-form structure is then as shown in Figure S13.1-2

(iii) Since the unit-sample response is symmetrical the filter is in fact linear phase. The
linear phase form is as shown in Figure S13.1-3.
Figure S13.l-3
(iv) The coefficients in the frequency-sampling structure are the samples of the frequency
response equally spaced in frequency. To evaluate these frequency samples it is
convenient to rewrite H(z) in the form

Because of the conjugate symmetry of the Fourier transform,
The frequency sampling structure, in the form of chalkboard (b) lecture 13, is then as shown
below:
Figure S13.1-4
In this form the sections involve complex coefficients. By utilizing the conjugate symmetry of the
frequency samples the network can be rearranged in terms of second-order sections with real

Specifically the transfer function corresponding to the recursive part of the network of Figure
S13.1-4 is given by
The terms paired in brackets are complex conjugates, i.e.,
Continuing these complex conjugate terms, G(z) can be expressed as
leading to the structure shown in Figure S13.1-5.

Figure S13.1-5 Solution 13.2 (a) The form of the desired transfer function is
easily obtained by expressing h(n) in the form

(b) The form of the transfer function for the coupled form network is (see Problem 11.3)
(c) For the direct form filter since the coefficient b is unity, it is not affected by quantization. For
the coefficient a1 , the quantized value is

In comparing this with the desired H(z) we note that since the co efficient of z-
2 is not unity, the resulting unit-sample response will not be of the desired
form. In particular, the unit-sample response will be of the form of a damped
sinusoidal sequence, corresponding to the fact that the poles of the coupled
form system with quantized coefficients have moved inside the unit circle. In
contrast, the poles of the direct form system with quantized coefficients remain
on the unit circle but are displaced in angle. These results are of course
consistent with the differences in the quantization grids for the two structures,
as illustrated in Figures 6.49 and 6.51 of the text.

The summation in the above expression is real. Let us assume for convenience that it is also
positive. If it is not, then for those values of k for which it is negative, an additional phase of 7 will
be added. Then, with this assumption,

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