Engineering Question

ADVANCED DIGITAL COMMUNICATIONS 7CCEMADC COURSEWORK There are 3 Questions, answer all. Detailed answers, calculations and codes (included in the written answers in pdf format) are required. Upload clearly scanned copies of your written answers by the deadline, as indicated on Keats. Dr. M. R. Nakhai, Department of Engineering, March 2024 Question 1 – Infinite-length design for channel equalisation [30 marks]: 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) = 1 𝑡 𝑡−𝑇 [sinc ( ) + 0.4sinc ( ) ], 𝑇 𝑇 √𝑇 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. 1 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). 1 𝑡 The transmit filter (basis function) is given as 𝜑(𝑡) = √𝑇 sinc (𝑇), where 𝑇 is the symbol period. The noise variance is 𝜎 2 = 0.125, the average transmit energy per dimension is ℰ̅𝑥 = 1. An oversampling factor of 𝑙 = 1 is assumed. The impulse 1 𝑡 response of the anti-aliasing filter at the receiver is given as ℎ𝑎𝑎 (𝑡) = 𝑇 sinc (𝑇), i.e., flat in the frequency domain with a gain of one. Hint: sinc ( 𝑡+𝑚𝑇 𝑇 𝑡+𝑛𝑇 ) ∗ sinc ( 𝑇 ) = 𝑇sinc ( 𝑡+(𝑚+𝑛)𝑇 𝑇 ), 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. 2 d) [4 marks] Find the resulting error variance 𝜎̅𝑀𝑀𝑆𝐸−𝐿𝐸 for your design in part (c). e) [3 marks] Compute the unbiased SNR for the MMSE-LE. 2 Question 3 – Finite-length design and evaluations with oversampling and polyphase channel modelling (requires MATLAB programming) [40 marks]: 1 Consider a filtered AWGN channel with impulse response ℎ(𝑡) = 𝑇 frequency response (Fourier transform of ℎ(𝑡)): 3𝜋 𝐻(𝑓) = 10 𝑒 −6𝜋(10 −7 )|𝑓| 1 107 𝑡 1+( 3 )2 and the . 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 𝛼 = 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 antialiasing 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) −7 3𝜋 in baseband for which you will need to find 𝐻(𝑓 + 𝑓𝑐 ) = 10 𝑒 −6𝜋(10 )|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 3 𝑇 (Project 1). With 𝑙 = 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., 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): 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. √𝑄(𝑓), as the continuoustime Fourier transform of 𝜑𝑟𝑐 (𝑡). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Form the continuous-time channel baseband spectrum, i.e., 𝐻(𝑓 + 𝑓𝑐 ) = 3𝜋 −6𝜋(10−7 )|𝑓+𝑓 | 𝑐 𝑒 10 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Form the continuous-time equivalent channel pulse response for finitelength 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 𝑇 sequence 𝑝𝑘 = 𝑝 (𝑘 2). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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: Question continues next page 4 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 ⋯ 𝒑𝝂 ]. d) [3 marks] Calculate channel power gain ‖𝑝‖2 in dB. 1 𝑇 2 2 2 Hint: Use ‖𝑝‖2 = ∑𝑘 |𝑝(𝑘 )| 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 . 5 Graph for the Q-Function Values Vertical axis: Horizontal axis: 2 ∞ −𝑢 𝑄(𝑥) = 2𝜋 ∫𝑥 𝑒 2 𝑑𝑢 √ 1 20log10 (𝑥) 6 7

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