EE2004: Digital Signal Processing

Tutorial-8
(April 28, 2025)
1. Consider the causal sequence x [n] = (−0.5)n u[n], with a z-transform given by X (z).
(a) Determine the inverse z-transform of X (z3 ) without computing X (z).
(b) Determine the inverse z-transform of (1 + z−2 ) X (z3 ) without computing X (z).
2. A discrete-time causal LTI system has the system function
(1 + 0.2z−1 )(1 − 9z−2 )
H (z) =
(1 + 0.81z−2 )
(a) Is the system stable?
(b) Find expressions for a minimum-phase system H1 (z) and an all-pass system Hap (z)
such that
H (z) = H1 (z) Hap (z)

3. The LTI systems H1 (e jω ) and H2 (e jω ) are generalized linear-phase systems. Which, if

any, of the following systems also must be generalized linear-phase systems?
(a) G1 (e jω ) = H1 (e jω ) + H2 (e jω )
(b) G2 (e jω ) = H1 (e jω ) H2 (e jω )
1
(c) G3 (e jω ) = 2π jθ j(ω −θ ) ) dθ
Rπ
−π H1 ( e ) H2 ( e

4. (a) A minimum-phase system has system function Hmin (z) such that
Hmin (z) Hap (z) = Hlin (z),
where Hap (z) is an all-pass system function and Hlin (z) is a causal generalized
linear-phase system. What does this information tell you about the zeros of Hmin (z)?
(b) A generalized linear-phase FIR system has an impulse response with real values
and h[n] = 0 for n < 0 and for n ≥ 8, and h[n] = −h[7 − n]. The system function
of this system has a zero at z = 0.8e jπ/4 and another zero at z = −2. What is H (z)?
5. The group delay of a filter is a measure of the average time delay of the filter as a
function of frequency. The group delay is defined as the negative first derivative of
the filter’s phase response. Determine the group delay for 0 < ω < π for each of the
following sequences:
(a) 
n − 1, 1 ≤ n ≤ 5
x1 [n] = 9 − n, 5 < n ≤ 9
0, otherwise


(b)
  | n −1|   | n |
1 1
x2 [ n ] = +
2 2

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 Practice Problems
6. A causal LTI system has impulse response h[n] for which the Z transform is

1 + z −1
H (z) =
(1 − 0.5z−1 )(1 + 0.25z−1 )
a) What is the ROC of H(z)?
b) Is the system stable?
c) Find the Z Transform X(z) of an input x[n] that will produce the output

1 4
y[n] = − (−0.25)n u[n] − 2n u[−n − 1]
3 3
d) Find the impulse response h[n] of the system.

7. When the input to an LTI system is

1
x [n] = ( )n u[n] + 2n u[−n − 1]
3
the corresponding output is
1 2
y [ n ] = 5( ) n u [ n ] − 5( ) n u [ n ]
3 3
a) Find the system function H(z) along with the ROC.
b) Find the impulse response of the system.
c) Find a difference equation that is satisfied by the given input and output.
d) Is the system stable ? Is the system causal?

8. Let y[n] is generated from x [n] as:

n
y[n] = ∑ kx [k ]
k=−∞
(a) Show that y[n] satisfies:

y[n] − y[n − 1] = nx [n]

and also
−z2 dX (z)
Y (z) =
z − 1 dz
where X (z) and Y (z) are the z-transforms of x [n] and y[n] respectively.
(b) Using this result find the z-transform of:
n  k
1
y[n] = ∑ k n≥0
k =0
3

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 9. Consider a causal LTI system with transfer function

1 − a −1 z −1
H (Z) =
1 − az−1
(a) Write the input-output difference equation for this system.
(b) For what values a is the system stable?
(c) For a = 0.5 , plot the pole-zero diagram and shade the region of convergence.
(d) Find the impulse response h[n] for the system.
1
10. If H ( Z ) = and h[n] = A1 α1n u[n] + A2 α2n u[n], determine the values of A1 , A2 , α1
1− 14 z−2
and α2
11. Determine the z transform and Region of Convergence (ROC) of each of the following
sequences. Express all sums in closed form, α can be complex.
(a) x a [n] = α|n| , 0 < |α| < 1.

1, 0 ≤ n ≤ N − 1,
xb [n] =
(b) 0, otherwise.
n, 0 ≤ n ≤ N,
(
(c)
xc [n] = 2N − n, N + 1 ≤ n ≤ 2N,
0, otherwise.
12. For each of the following system functions Hk (z), specify a minimum-phase system
function Hmin (z) such that the frequency-response magnitudes of the two systems are
equal, i.e.,
| Hk (e jω )| = | Hmin (e jω )|.
1−2z−1
(a) H1 (z) =
1+ 13 z−1
(1+3z−1 )(1− 21 z−1 )
(b) H2 (z) =
(z−1 )(1+ 13 z−1 )
(1−3z−1 )(1− 41 z−1 )
(c) H3 (z) =
(1− 34 z−1 )(1− 43 z−1 )

13. Consider the class of FIR filters that have h[n] real, h[n] = 0 for n < 0 and n > M, and
one of the following symmetry properties:

Symmetric: h[n] = h[ M − n]
Antisymmetric: h[n] = −h[ M − n]
All filters in this class have generalized linear phase, i.e., have frequency response of
the form
H (e jω ) = A(e jω )e− jαω + jβ ,
where A(e jω ) is a real function of ω, α is a real constant, and β is a real constant.
For the following table, show that A(e jω ) has the indicated form, and find the values
of α and β.

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Filter length (M+1) Form of A(e jω ) Type Symmetry α β

M/2
M/2 ∑n=0 a[n] cos(ωn) I Symmetric Odd
  
( M+1)/2 1
( M + 1)/2 ∑ n =1 b[n] cos ω n− 2 II Symmetric Even

M/2
M/2 ∑n=1 c[n] sin(ωn) III Antisymmetric Odd
  
( M+1)/2
( M + 1)/2 ∑ n =1 d[n] sin ω n − 21 IV Antisymmetric Even

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