1. Two different signals X 1 ( z ) and X 2 ( z ) are transmitted through independent channels in
close proximity so that they are received linearly mixed as Y1 ( z )  AX 1 ( z )  BX 2 ( z ) and
Y2 ( z )  AX 1 ( z )  BX 2 ( z ) and modelled as shown in the following Figure 1.
The mixing matrix is A  D  1, B  H 12 ( z ) and C  H 21 ( z ) with H 12 ( z ) and H 21 ( z )
unknown FIR transfer functions.
The signals Y1 ( z ) and Y2 ( z ) are further processed by a linear system which has a mixing
B   G12 ( z ) and C   G 21 ( z )
matrix A  D   1, B   G12 ( z ) and C   G21 ( z ) with
unknown transfer functions, to produce two new signals U 1 ( z ) and U 2 ( z ) .
(a) Determine the relationship between the functions H 12 ( z ), H 21 ( z ), G12 ( z ) and G 21 ( z ) so
that each of U 1 ( z ) and U 2 ( z ) is a function of only one of the original signals X 1 ( z ) and
X 2 ( z ) . Determine the two alternative solutions possible and the associated
proportionality transfer functions in terms of the mixing parameters above.
(b) It is not known a priori whether the mixing channel transfer functions are minimum phase
or not. Comment on possible limitations of one of the alternative solutions.
X 1 ( z)
X 2 ( z)
Y1 ( z )
A
C

B
D

Y2 ( z )
Figure 1
Answer
The final mixing matrix is given by the following relationship:
 A B   A B    1 B   1 B  1  BC  B  B  
C D  C  D   C 1  C  1    C  C  1  B C 


 

 

(a) The first possible design is
U 1 ( z )  F1 ( z ) X 1 ( z ) and U 2 ( z )  F2 ( z ) X 2 ( z )
B  B   0  B    B  G12 ( z )  H 12 ( z )
C  C   0  C   C  G 21 ( z )  H 21 ( z )
The second possible design is
1  BC   1  BC  1  H 12 ( z ) H 21 ( z )  1  B C
U 1 ( z )  (1  H 12 ( z ) H 21 ( z )) X 1 ( z )
U 2 ( z )  (1  H 12 ( z ) H 21 ( z )) X 2 ( z )
U 1 ( z )  K1 ( z ) X 2 ( z ) and U 2 ( z )  K 2 ( z ) X 1 ( z )
1
1
 G21 ( z ) 
B
H 12 ( z )
1
1
1  B C  0  B     G12 ( z ) 
C
H 21 ( z )
H ( z ) H 21 ( z )  1
1
C  C   H 21 ( z ) 
 12
 B  B
H 12 ( z )
H 12 ( z )
( H ( z ) H 21 ( z )  1)
U 1 ( z )  12
X 2 ( z)
H 12 ( z )
( H ( z ) H 21 ( z )  1)
U 2 ( z )  12
X 1 ( z)
H 12 ( z )
1  BC   0  C   
(b) In the second case a problem arises if one or both functions H 12 ( z ), H 21 ( z ) have zeros
outside the unit circle. In that case the one or both filters G12 ( z ), G 21 ( z ) will be unstable.
2. (a) Show that the analogue transfer function
a
(1)
,a 0
sa
has a lowpass magnitude response with a monotonically decreasing magnitude response
with H 1 ( j 0)  1 and H 1 ( j)  0 . Determine the 3-dB cut-off frequency  c at which
the gain response is 3 dB below the maximum value of 0 dB at   0 .
H 1 ( s) 
(b) Show that the analogue transfer function
s
(2)
,a0
sa
has a highpass magnitude response with a monotonically increasing magnitude response
with H 1 ( j 0)  0 and H 1 ( j)  1 . Determine the 3-dB cut-off frequency  c at which
the gain response is 3 dB below the maximum value of 0 dB at    .
H 2 ( s) 
(c) The lowpass transfer function H 1 ( s ) of equation (1) and the highpass transfer function of
equation (2) can be expressed in the form
1
1
H 1 (s)  A1 (s)  A2 (s), H 2 (s)  A1 (s)  A2 (s)
2
2
where A1 ( s ) and A2 ( s ) are stable analogue allpass transfer functions. Determine A1 ( s )
and A2 ( s ) .
Answer
(a) H 1 ( s ) 
a
a
a2
2
 H 1 ( j ) 
 H 1 ( j )  2
.
sa
j  a
a  2
H 1 ( j)
2
is
a
monotonically decreasing function with H 1 ( j 0)  1 and H 1 ( j)  0 . Let the 3-dB cut2
off frequency be given by  c . Hence, H 1 ( j c ) 
a2
1
  c  a .
2
2
a  c 2
(b) Similarly, H 2 ( s ) 
j
s
2
2
2
 H 2 ( j ) 
 H 2 ( j )  2
. H 2 ( j) is a
2
sa
j  a
a 
monotonically increasing function with H 2 ( j 0)  0 and H 2 ( j)  1 . Let the 3-dB cut-
 c2
2
off frequency be given by  c . Hence, H 2 ( j c ) 
(c) H 1 ( s) 
a 
2

1
 c  a .
2
a
1  ( s  a) 
s
1  ( s  a) 
 1 
 1 
 and H 2 ( s) 
.
s  a 2  ( s  a) 
s  a 2  ( s  a) 
s 1
.
Since
sa
A1 ( s) and A2 ( s) are both allpass transfer functions.
Hence,
 c2
A1 ( j c )  1 and A2 ( j c )  1 ,
A1 (s)  1 and A2 (s) 
3. (a) In an audio application the structure in Figure 3 is used as a “reverberator” to reproduce
attenuated versions of an impulse applied as input. Determine its transfer function and
show that it is allpass.
(b) Propose an alternative canonic or non-canonic signal flow graph with one multiplier that
realises the transfer function. Explain every step in your answer.
(c) Determine the impulse response of the structure knowing that after 800 msecs the
impulse response has an absolute value equal to the 1% of its value at the instant n  1 .
The signals are sampled at 10 kHz.


x (n)
y (n)


1 2

z 1


U (z )

Figure 3
Answer
(a) Let the output of the intermediate adder be U (z ) . Then, in the z  transform domain we
can write
X ( z)
U ( z )  X ( z )  z 1U ( z )  U ( z ) 
1  z 1
At the output adder we have
X ( z)
Y ( z )  X ( z )  (1   2 ) z 1U ( z )  X ( z )  (1   2 ) z 1
1  z 1
or
Y ( z)
1
    2 z 1  z 1   2 z 1 z 1  
H ( z) 
   (1   2 ) z 1


X ( z)
1  z 1
1  z 1
1  z 1
To show that H (z ) is allpass we can write it as
z 1 (1  z ) z 1 (1  z 1 ) 

 H ( z)
1  z 1
1  z 1
(Notice that (1  z 1 )   (1  z 1 ) )
H ( z) 
From H ( z ) 
z 1
1
Y ( z ) z 1  

we can write Y ( z )(1  z 1 )  X ( z )( z 1   ) and hence
1
X ( z ) 1  z
Y ( z )  z 1Y ( z )  X ( z )( z 1   )
For a single multiplier realisation we can write the above equation as
Y ( z )  z 1 X ( z )   [ z 1Y ( z )  X ( z )]
(b) At this stage we can have various signal flow graphs depending on any additional
constraints to be taken into consideration.
A direct non-canonic realisation is to generate separately the components on the right
hand side of equation Y ( z )  z 1 X ( z )   [ z 1Y ( z )  X ( z )] . With the input and the output
nodes defined we have for the second term

X (z )
Y (z )
+
-
+
z 1
while the first term of the same equation is directly implemented as
X (z )
Y (z )
z
1
On combining the two signal flow graphs we obtain
+
X (z )
Y (z )
+
z
1
+

+
-
+
z 1
(c) The impulse response h(n) is given by
 z 1   
z 1 
1   
h(n)   1 






1
1
1  z 1 
1   z 
1   z
  n   n 1
n 1

h ( n )   
n0
0
n0

 n 1 (1   2 )

h ( n )   
0

n 1
n0
n0
Since the sampling frequency is 10kHz duration of 800 msecs is equivalent to 8000
samples. The impulse response is
1
h(n)   8000 (1   2 ) 
(1   2 )    0.9994
100
4. (a) The transfer function of an ideal real coefficient lowpass filter H LP (z ) has cutoff
frequency  p and impulse response h LP (n) . Show that:
(i) that H LP ( z ) is a highpass filter and of cutoff frequency    p .
(ii) Find an expression for the impulse response hHP (n) of this highpass filter in terms of
the impulse response h LP (n) of the original lowpass filter.
(b) It is proposed to use a given stable lowpass digital filter transfer function H LP (z ) of
impulse response h LP (n) to derive a bandpass transfer function G(z ) centred at  0
according to the formula:
G( z )  H LP ( ze j0 )  H LP ( ze  j0 ) .
(i) Determine the impulse response g (n) of the bandpass filter in terms of the impulse
response h(n) of the lowpass filter and the centre frequency  0 .
(ii) Show that the signal flow graph in the figure below realises the transfer function
G(z ) .
Answer
(a)
(i)
Replacing z with  z is a z  plane rotation by 180 0 . Hence, the passband
  p     p will be rotated to be around the point z  1  e j that corresponds
to frequency    , i.e.,    p       p . This is a highpass filter response.
(ii)
The impulse response hHP (n) is the inverse z  transform of the transfer function
H LP ( z ) .
n  
n  
n  
n  
H HP ( z )  H LP ( z )   hLP (n)( z ) n   hLP (n)(1)  n z  n
n  
  [hLP (n)(1) n ]z  n  hHP (n)  hLP (n)(1) n
n  
(b)
n  
(i)
G( z )  H LP ( ze j0 )  H LP ( ze  j0 )   [hLP (n)( ze j0 ) n  hLP (n)( ze  j0 ) n ]
n  
n  
n  
n  
n  
n  
n  
  [hLP (n)( ze j0 ) n  hLP (n)( ze  j0 ) n ]   [hLP (n)(e j0 ) n  hLP (n)(e  j0 ) n ]z n
  [hLP (n)(e j 0 )  n  hLP (n)(e  j 0 )  n ]z  n   hLP (n)[(e j 0 )  n  (e  j 0 )  n ]z  n
n  
n  
n  
  hLP (n)[(e
n  
(ii)
j 0  n
)
 (e
 j 0  n
) ]z
n
n  
  2hLP (n) cos( 0 n) z  n
n  
Thus, g (n)  2hLP (n) cos( 0 n)
The output from the lowpass filter of the top part of the structure is the convolution
between the impulse response of the filter hLP (n) and the signal 2 x(n) cos( 0 n)
k  
which is
 2hLP (n  k )x(k ) cos(0 k ) . The output from the lowpass filter of the
k  
bottom part of the structure is the convolution between the impulse response of the
k  
filter hLP (n) and the signal 2 x(n) sin( 0 n) which is  2hLP (n  k )x(k ) sin(0 k ) .
k  
Therefore,
k  
k  
y (n)  cos( 0 n)  2hLP (n  k )x(k ) cos( 0 k )  sin( 0 n)  2hLP (n  k )x(k ) sin( 0 k )
k  
k  
k  
k  
k  
k  
  2hLP (n  k )x(k ) cos( 0 n) cos( 0 k )   2hLP (n  k )x(k ) sin( 0 n) sin( 0 k )
k  
  2hLP (n  k )x(k )[cos( 0 n) cos( 0 k )  sin( 0 n) sin( 0 k )]
k  
k  
  2hLP (n  k ) cos 0 (n  k ) x(k )  g (n)  x(n)
k  
Therefore, the above structure realises the filter
5. (a) Prove that the up-sampler and the down-sampler are linear but tine-varying discrete time
systems.
(b) Prove that down sampling by a factor of M followed by filtering with a filter with
transfer function H (z ) is equivalent with filtering first with a filter with transfer
function H ( z M ) followed by down sampling by a factor of M .
Answer
Bookwork