Digital filters - Continuous and Discrete Time Signals and Systems - page 663

14.14.

14.16 An allpass filter has a constant gain for all frequencies, i.e. H ( Ω ) = 1.
(i) Show that the transfer functions

− 1 − 1 + z − 2

α 1 + z

H 2 ( z ) = α 1 α 2 + α 1 z H 1 ( z ) = and

1 + α 1 z − 1 1 + α 1 z − 1 + α 2 z − 2

represent allpass filters.
(ii) Sketch the flow graph for the first-order allpass filter H 1 ( z ), which
uses a single scalar multiplier.
(iii) Sketch the flow graph for the second-order allpass filter H 2 ( z ) with
only two scalar multipliers. There is no restriction on the number
of unit delay elements or two-input adders in each case.

14.17 The impulse response of an LTID system is given by

α k 0 ≤ k ≤ 9

h [ k ] =

0 elsewhere .

(i) Draw the flow graph for the above LTID system with no feedback
paths.
(ii) The z-transfer function for the above impulse response is given by

1 − α 10 z − 10
1 − α z − 1

H ( z ) = .

Draw the flow graph of the IIR system specified by this transfer
function.
(iii) Compare the two implementations with respect to the number of
delays, scalar multipliers and two-input adders.

14.18 Implement the filter with transfer function given by

H ( z ) = 0 . 4 − 0 . 8 z − 1 + 0 . 4 z − 2

with finite-precision arithmetic given by

( − 1) s (0 + 0 . significand) ,

where the significand represents the decimal fraction of the coefficients
and is limited to 3 bits with 1 bit allocated for the sign. Compare the
magnitude response of the original filter with the magnitude response of
the filter implemented with finite-precision representation.