Successive quadrature amplitude modulation (QAM) symbols are transmitted through a filtered additive white Gaussian noise channel with pulse response p(t), given as:

p(t) = sinc  + 0.4sinc ,

Where T  is the symbol period. The matched filter bound on signal to noise ratio (SNR) for this transmission is SNRMFB  =  16 dB. The sampling is performed at symbol rate.

a)  [10 marks] Design a minimum mean square error linear equaliser (MMSE-LE) by finding its filter transform, i.e., WMMSE-LE (D), and the resulting unbiased SNR, i.e., SNRMMSE-LE,U  for this transmission system.

b)  [10 marks] Design a minimum mean square error decision feedback equaliser (MMSE-DFE) by finding its feed-forward and feedback filter transforms, i.e., WMMSE-DFE (D)   and   BMMSE-DFE (D),  and  the  resulting  unbiased  SNR,  i.e., SNRMMSE-DFE,U for this transmission system.

c)  [4 marks] Draw the detailed block diagram of your design in part (b).

d)  [6 marks] This channel is used to transmit successive symbols drawn from a 4-QAM (quadrature amplitude modulation) constellation.

Find the approximate probability of symbol error pe  for one of the equalisers of your design that achieves a higher SNR. Also explain why the chosen design has achieved a higher SNR.

Question 2 - Finite-length design [30 marks]:

Consider a filtered additive white Gaussian noise (AWGN) channel with an impulse response modelled as:

ℎ(t) = δ(t) − 0.5δ(t − T).

The transmit filter (basis function) is given as φ(t) =  sinc , where T  is the

symbol period. The noise variance is  σ 2   = 0.125, the average transmit energy per

dimension is  ℰ(̅)x   = 1. An oversampling factor of l  =  1  is assumed. The impulse

response of the anti-aliasing filter at the receiver is given as ℎaa (t) = i.e., flat in the frequency domain with a gain of one.

 sinc ,

Hint: sinc  ∗ sinc  = Tsinc , where ∗  means convolution.

Important attention: The objective of this question is to explore the calculations   of the formulations in lecture notes in details. You may use MATLAB for any matrix manipulations required for completing your solution. You may also validate your detailed solutions and answers with the MATLAB DFE programme in KEATS, after you have solved the problem and written down your solutions in details.

a)  [8 marks] Find the pulse response p(t) of the channel for finite-length design and the power gain ‖p‖2 , corresponding to the discrete-time channel.

b)  [3 marks] Find the signal-to-noise ratio (SNR) matched filter bound, i.e., SNR MFB  (in dB).

c)  [12 marks] Design a 3 tap finite impulse response minimum mean square error linear equaliser (FIR MMSE-LE) for delay Δ = 0.

d)  [4 marks] Find the resulting error variance̅(σ)M(2)MSE−LE   for your design in part (c).

e)  [3 marks] Compute the unbiased SNR for the MMSE-LE.

Question 3 - Finite-length design and evaluations with oversampling and polyphase channel modelling (requires MATLAB programming) [40 marks]:

Consider a filtered AWGN channel with impulse response ℎ(t) =  and the frequency response (Fourier transform of ℎ(t)):

H(f) =  e −6π(10−7)|f| .

QAM transmission with symbol rate of 1 MHz and carrier frequency fc   = 500 KHz is used on this channel with oversampling factor of 2. The transmission system is shown in Figure 1:

Figure 1. Detailed transmission channel

As shown on Figure 1, 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 l  = 2. Square root raised cosine pulse φrc (t) with 10% of excess bandwidth (i.e., roll-off factor a  = 0. 1) is used as the transmit basis function. An ideal anti-aliasing filter with frequency response as characterised on Figure 1 is   applied at the receiver input.

A MATLAB program provided on KEATS (Project 1) can be used to implement φrc (t) . According to the lecture notes the complex pulse response is modelled as the cascade of the transmit basis function φrc (t), baseband equivalent of the channel impulse response, ℎbb (t), and the impulse response of the ideal anti- aliasing filter, ℎaa (t):

p(t) = φrc (t)  ∗ ℎbb (t) ∗ ℎaa (t) .

a)  [18 marks] Using MATLAB programming, find the complex discrete-time

pulse response samples and write down as your answer for part (a) only the 8 samples around the peak value (i.e., centred around time origin t  = 0  , with time zero included), as the truncated pulse response for practical design.

Hint: Using MATLAB, first find the continuous-time Fourier transform of p(t) in baseband for which you will need to find H(f + fc ) =  e −6π(10−7)|f+fc| .

You will also need square root raised cosine pulse in frequency domain, i.e., √Q(f) in lecture notes, whose MATLAB program can be found in KEATS

(Project 1). With l  = 2 , your sampling period will be . 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., N-point IFFT in MATLAB, you will be able to find the samples of the pulse response. To avoid aliasing in discrete-time domain, use N  large enough, i.e., N  = 1024. After obtaining N  discrete-time samples of pulse response, truncate it around the peak of the pulse response, i.e., around the time origin, and keep 8 samples.

Here is a minimum programming steps, suggested for part (a):

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Form raise cosine pulse in frequency-domain, i.e. √Q(f), as the continuous- time Fourier transform of φrc (t) .

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Form the continuous-time channel baseband spectrum, i.e., H(f + fc ) =  e −6π(10−7)|f+fc|

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Form the continuous-time equivalent channel pulse response for finite- length in frequency-domain, i.e., P(f)  as the continuous-time Fourier

transform of          p(t) = φrc (t)  ∗ ℎbb (t) ∗ ℎaa (t).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Form the discrete-time Fourier-transform (DTFT) of the discrete-time sequence pk   = p (k  .

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Use the N-point IFFT command to find the discrete-time samples of the pulse response pk .

Use the ifftshift command to centre time samples about the time origin

Record 8 time samples of pk   (4 samples to the left of the time origin, sample at the origin and 3 samples to the right of the time origin).

b)  [2 marks] Since you have truncated pk   and kept only 8 samples as in part   (a), compute the error incurred in doing so, i.e., as a result of ignoring the channel power gain/attenuation.

Hint: First calculate sum of the magnitude squares of the kept part of pk divided by sum of the magnitude squares of all samples of pk .

Continue Question 3 with the following remaining parts, which are independent from Parts (a) and (b) of this question:

The discrete-time complex pulse response at carrier frequency fc   = 600 KHz, the   symbol rate of 1 MHz for QAM transmission through the transmission system shown and detailed in Figure 1 with oversampling factor of l  = 2  is given as:

P(D) = (0.0212 − j0.0478)D−4  + (0.0879 + j0.0295)D−3  + (−0.0428 + j0.0945)D−2 + (−0.2085 − j0.1372)D−1  + 0.364 + (−0.2085 + j0.1372)D

+ (−0.0428 − j0.0945)D2  + (0.0879 − j0.0295)D3

c)  [5 marks] After applying delay to make pk   causal, express the obtained  samples as causal 2-tuple representation of the polyphase channel pulse response, i.e., calculate the matrix p =  [p0  p1   ⋯  pv] .

d)  [3 marks] Calculate channel power gain ‖p‖2   in dB.

Hint: Use ‖p‖2  =  ∑k |p(k ) |2

e)  [2 marks] Calculate signal-to-noise ratio matched filter bound.

f)  [5 marks] Using the DFE programme in KEATS, design an FIR MMSE-LE with 10 taps and find the filter coefficients and the corresponding achievable

data rate at probability of symbol error of 10−6 .

g)  [5 marks] Using the DFE programme in KEATS, design an FIR MMSE-DFE with 10 feedforward taps and 4 feedback taps and find the filter coefficients and corresponding achievable data rate at probability of symbol error of 10−6 .