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Advanced Digital Communications 7CCEMADC
Coursework
There are 4 Questions, answer all.
Detailed answers, calculations, diagrams and codes,
included in the written answers in pdf format, are
required.
Upload clearly scanned copy of your written answers by the
deadline, as indicated on KEATS.
Dr. M. R. Nakhai, Department of Engineering,
March 2025
7CCEMADC
Question one (Finite-length Design) [25 marks]
Figure 1: Finite-length transmission system design
The sampled overall pulse response, i.e., ( ) = ( ) ( ) ( ), of a 2-ray
transmission system in Figure 1, is given by
= + 0.8 1
,

The oversampling factor is unity, i.e., = 1, the sampling has been done at symbol
rate. Quadrature amplitude modulation (QAM) symbols are transmitted with average
energy per dimension of . The noise variance is
2 =
0
2
= 0.164.
Use MATLAB and follow direct formulations step-by-step for answering the
following parts of this question.
Solutions using the DFE program in parts (a) and (b) are not acceptable and will
not be marked for this Question. But you can use the DFE program in parts (a)
and (b) to double check your MATLAB-assisted numerical solutions, for yourself.
a) Following the step-by-step finite-length formulations, design an FIR minimum
mean square error-linear equaliser (MMSE-LE) filter, by calculating the
discrete-time impulse response with 4 coefficients, i.e., = 4, and the
resulting signal to noise ratio (SNR), i.e.,
SNR MMSE-LE,U
, with an overall delay
factor of Δ = 2.
(7 marks)
b) Repeat part (a), for a minimum mean square error-decision feedback
equaliser (MMSE-DFE) with = 3 and one feedback tap, i.e., = 1, and
find the feedforward and the feedback filters’ coefficients as well as the
achievable by directly following the finite-length formulations
with an overall delay factor of Δ = 2.
(7 marks)
c) Using the DFE program, find the optimum delay that minimises mean-square
of the error signal in part (b).
(2 marks)
d) Complexity analysis and implementation planning for your MMSE-DFE design
in part (b):
I. Draw the detailed structure of your MMSE-DFE design, which includes
detailed structures of the feedforward and the feedback digital filters.
(2 marks)
Question one continues next page
=1 x E
SNR MMSE-DFE,U
7CCEMADC
II. Suppose successive 2-PAM (pulse amplitude modulation) symbols are
intended to be transmitted at a data rate of 100 Mbps using your
design in part (b). You are asked to order a suitable digital signal
processor (DSP) from Intel for real-time implementation of your MMSE DFE. What is your order for the DSP in terms of minimum required
speed in million instructions per second (MIPS)
(3 marks)
III. Repeat part (d_II) for your MMSE-LE design in part (a).
(2 marks)
e) Compare your results in parts (a) and (b) in terms of complexity of
implementation and the achievable SNR after equalisation using MMSE-LE and
MMSE-DFE and conclude.
(2 marks)
See Next Page
7CCEMADC
Question two (Finite-length Design) [25 marks]
Figure 2: Finite-length transmission system design
For the transmission system in Figure 2, the two-ray channel impulse response in the
baseband is given as
( ) = ( ) + 0.8 ( ).
The impulse responses of the transmit filter, i.e., the basis function ( ), and the
anti-aliasing filter, i.e., ( ), respectively, are:
( ) =
1
√
sinc (

) ,
( ) =
√

sinc (

),
where is the oversampling factor and is the symbol period. Quadrature
amplitude modulation (QAM) symbols are transmitted with an oversampling factor
of = 2 and an average energy per dimension of . The noise variance is.

2 =
0
2
= 0.164.
a) Find the expression for the overall channel pulse response ( ) for the finite length design.
(2 marks)
b) Find the discrete-time expression for the oversampled ( ), i.e., , with an
oversampling factor of = 2.
(2 marks)
c) Using MATLAB, calculate the oversampled coefficients of ( ) with = 2,
keeping an overall duration of 5 of ( ), i.e., when 2.5 ≤ ≤ 2.5 , and
ignoring the rest of the energy of ( ).
Hint: sinc( ) =
sin

( )
(2 marks)
d) Using D-transform, express the resulting discrete-time signal in part (c).
(1 mark)
e) Find the causal expression for the oversampled samples in discrete-time
domain.
(1 mark)
Question Continues Next Page
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=1 x E
7CCEMADC
f) Find the matrix expression for the causal samples in part (e), i.e., the
expression [ 0 1
], where
is a × 1 vector of respective
coefficients. (1 mark)
g) Using the DFE program, design a minimum mean square error-linear equaliser
(MMSE-LE) digital filter (with fractionally-spaced delay line) by finding its
discrete-time impulse response coefficients and the resulting unbiased
signal to noise ratio (SNR), i.e.,
SNR MMSE-LE,U
, with = 4 (i.e., number of
coefficients corresponding to the symbol spaced delay line of the MMSE-LE
filter). Use Δ = 4 for the overall delay factor.
(2 marks)
h) Repeat part (g) for a minimum mean square error-decision feedback equaliser
(MMSE-DFE) with = 3 and one feedback tap, i.e., = 1 and record the
corresponding feedforward and feedback filters’ coefficients as well as the
achievable . Use Δ = 4 for the overall delay factor.
(2 marks)
i) Complexity analysis and implementation planning for your designs:
i. Draw the detailed structure of your MMSE-DFE design, which includes
detailed structures of the feedforward and the feedback digital filters
with all coefficients written on the figure.
(3 marks)

ii. Suppose successive 2-PAM (pulse amplitude modulation) symbols are
intended to be transmitted at a data rate of 100 Mbps using your MMSE DFE design in part (h). You are asked to order a suitable digital signal
processor (DSP) from a company like Intel for real-time
implementation of your MMSE-DFE. What would be your order for the
DSP in terms of minimum required speed in million instructions per
second (MIPS)
(3 marks)
iii. Repeat part (i_II) for your MMSE-LE design in part (g).
(3 marks)
j) Compare your results in parts (g) and (h) in terms of the complexity of
implementation and the achievable SNR after equalisation using MMSE-LE and
MMSE-DFE and conclude.
(3 marks)
See Next Page
SNR MMSE-DFE,U
7CCEMADC
Question three – Finite-length design and evaluations with oversampling and
polyphase channel modelling (mini project requires MATLAB programming)
[25 marks]:
Consider a filtered AWGN channel with impulse response ( ) =
1 1
1+(
10
3
7
)
2
and the
frequency response (Fourier transform of ( )):
( ) =
3
10

6 (10 7)| |
.
QAM transmission with symbol rate of 1 MHz and carrier frequency = 650 KHz is
used on this channel with oversampling factor of 2. The transmission system is
shown in Figure 3:
Figure 3. Detailed transmission channel
As shown on Figure 3, the power spectral density of noise is 86.5 dBm/Hz and the
average transmit power is 1mW. The oversampling factor for the design of the
equaliser is = 2. Square root raised cosine pulse ( ) with 10% of excess
bandwidth (i.e., roll-off factor = 0.1) is used as the transmit basis function. An
ideal anti-aliasing filter with frequency response as characterised on Figure 3 is
applied at the receiver input.
A MATLAB program provided on KEATS (Project 1) can be used to implement
( ). According to the lecture notes the complex pulse response is modelled as
the cascade of the transmit basis function ( ), baseband equivalent of the
channel impulse response, ( ), and the impulse response of the ideal anti aliasing filter, ( ):
( ) = ( ) ( ) ( ).
a) Using MATLAB programming, find the complex discrete-time pulse response
samples and write down as your answer for part (a) only the 9 samples
around the peak value (i.e., centred around time origin = 0 , , =
[ 4,+4]), as the truncated pulse response for practical design.
(20 marks)
Question Continues Next Page
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Hint: Using MATLAB, first find the continuous-time Fourier transform of ( ) in
baseband for which you will need to find ( +
) =
3
10

6 (10 7)| + |
. You will
also need square root raised cosine pulse in frequency domain, i.e., √ ( ) in
lecture notes, whose MATLAB program can be found in KEATS (Project 1). With =
2, your sampling period will be
2
. Then you will be able to find the discrete-time
Fourier transform (DTFT) of the oversampled discrete-time pulse response. Then
using inverse DTFT, i.e., -point IFFT in MATLAB, you will be able to find the
samples of the pulse response. To avoid aliasing in discrete-time domain, use
large enough, i.e., = 1024. After obtaining discrete-time samples of pulse
response, truncate it around the peak of the pulse response, i.e., around the time
origin, and keep 9 samples.
Here is a minimum programming steps, suggested for part (a):
T = 1*10^(-6); % Symbol Period
fc = 5*10^(5); % Carrier Frequency
alpha = 0.1; % Excess Bandwidth
l=2; %oversampling factor
N = 1024; % Number of FFT points
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Form raise cosine pulse in frequency-domain, i.e. √ ( ), as the continuous time Fourier transform of ( ). [5 marks]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Form the continuous-time channel baseband spectrum, i.e., ( +
) =
3
10

6 (10 7)| + |
[3 marks]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Form the continuous-time equivalent channel pulse response for finite length in frequency-domain, i.e., ( ) as the continuous-time Fourier
transform of ( ) = ( ) ( ) ( ). [2 marks]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Form the discrete-time Fourier-transform (DTFT) of the discrete-time
sequence = (

2
). [2 marks]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Use the N-point IFFT command to find the discrete-time samples of the
pulse response .
Use the ifftshift command to centre time samples about the time origin
Record 9 time samples of (4 samples to the left of the time origin,
sample at the origin and 4 samples to the right of the time origin).
[8 marks]
b) Since you have truncated and kept only 9 samples as in part (a), compute
the percentage of the error incurred in doing so, i.e., as a result of ignoring
the remaining fraction of the channel power gain/attenuation.
(5 marks)
Hint: First calculate sum of the magnitude squares of the kept part of
divided by sum of the magnitude squares of all samples of .
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7CCEMADC
Question four (Infinite-length Design) [25 marks]
Successive quadrature amplitude modulation (QAM) symbols are transmitted through
a filtered additive white Gaussian noise (AWGN) channel with matched filter bound
signal to noise ratio (SNR) of SNRMFB = 50, in linear scale. When this channel is
equalised by a zero forcing – decision feedback equaliser (ZF-DFE) using a feedback
filter characterised by 1 ZF-DFE( ) = , an equalised SNR of SNRZF-DFE = 25, in
linear scale, can be achieved.
Assume, in linear scale, the ratio of
2 = 25, where and
2 are the average energy
of transmission per dimension and the variance of noise, respectively. A gap of
Γ = 8.8 dB at a target probability of symbol error of = 10 6
is assumed for this
transmission system.
a) Find the channel deterministic autocorrelation function ( ) and the
feedforward filter transform of the used ZF-DFE, i.e., ZF-DFE( ).
(6 marks)
For the following parts assume ( ) = 0.5
1 + 1 0.5 :
b) Find the feed-forward and the feedback filter transforms for a minimum mean
square error – decision feedback equaliser (MMSE-DFE), i.e., MMSE-DFE( ) and
MMSE-DFE( ), and compute the corresponding unbiased SNR, i.e.,
SNRMMSE-DFE,U, for this channel.
(6 marks)
c) Find the filter transform MMSE-LE( ) and the corresponding unbiased SNR,
i.e., SNRMMSE-LE,U, for a minimum mean square error – linear equaliser
(MMSE-LE) for this channel.
(6 marks)
d) By comparing your results in the previous parts, choose an equaliser that gives
you the highest data rate on this channel. Intuitively explain that on the basis
of what important fact this should be your choice.
(2 marks)
e) With an average transmission power of

= 20 dBm, where is the symbol
period, and the noise power spectral density of
2 = 60 dBm/Hz, find the
maximum achievable data rate on this channel, using your chosen equaliser
in part (d), with an integer number of bits per symbol at target probability of
symbol error = 10 6
. (Hint:
0 dBm =1mW
.)
(5 marks)
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Graph for the Q-Function Values
Vertical axis: ( ) =
1
√2
∫

2
2

∞
Horizontal axis: 20log10( )
Final Page