Question 1:

Given a signal x (t) sampled at 21 kHz, you are required to design the most efficient multi-rate filtering scheme using only linear phase FIR filters, with the following overall filtering specifications:

State All Assumptions. (Hint:  You  will  need  to  try  a  few  configurations  to determine the best.)

|H(f)|

4 kHz       5 kHz        Frequency

a.  Draw a block diagram of your overall scheme.

b.  For every FIR filter designed, state all the parameters used, namely,

i.     Passband frequency (Hz)

ii.     Transition band (Hz)

iii.     Pass-band ripple

iv.     Stop-band ripple

v.     Order of the filter (Method and Computation must be shown)

c.  Work  out  the  multiplication  rates  of  each  filter  and  hence  the  overall multiplication rate.

(60 marks)

Question 2:

P0(z)

P1(z)

PL−1(z)

Figure 2

Figure 2 shows an interpolator implemented using a polyphase structure with a commutator where the input sampling rate is Fs  and interpolation rate is L.

a.  Work out the output time-domain expression of y(m) in terms of the input   x(n) and polyphase sub-filters, pi (n), where Pi (z) = ∑n(∞)=0 pi (n)z−n ,     i = 0, 1, … , L − 1.

b.  Hence or otherwise work out Y(z) in terms of the input and polyphase sub- filters, where Y(z) = ∑m(∞)=0 y(m)z−m

(20 Marks)

Question 3:

Given a discrete-time signal x(n), investigate the type of filter to be used to achieve linear interpolation by a factor of 3 as shown in Figure 3.

a.  Derive the filter to be used to achieve linear interpolation?

b.  Compare the differences between the use of a low-pass brick filter and linear interpolation filter, in terms ofup-sampling accuracy. (State all assumptions)

12

10

8

6

4

2

0

-2

-4

x(n)

0                                0.5                                     1                                1.5                                     2                                2.5                                     3                                3.5                                     4

12

10

8

6

4

2

0

-2

-4

0                                                     2                                                     4                                                     6                                                     8                                                   10