matlab - Low-pass filter in carrier modulation and demodulation -

i'm designing project in array passed through quadrature amplitude modulation (qam) modulator, , carrier modulation, make playable sound() command, demodulate qam demodulation.

firstly, have used standard way of qam modulation:

m = 16; x = randint(5000, 1, m); y = modulate(modem.qammod(m), x); 

then, wrote own carrier modulation function:

function [out] = carriermodulation(x) fs = 16000; t = 1.0 / 4000; fc = 8000; q = real(x); = imag(x); t = 0:t:(size(x))*t; c1 = zeros(size(x), 1); c2 = zeros(size(x), 1); = 1:size(x)     c1(i) = i(i)*sin(2*pi*(fc)*t(i));     c2(i) = q(i)*sin(2*pi*fc*t(i) + pi/2); end  out = c1 + c2; 

no problem far. when done demodulation function, found result different original value (the qam modulator output).

function [out] = carrierdemodulation(x) fs = 16000; t = 1.0 / 4000; fc = 8000; t = 0:t:(size(x))*t; a1 = zeros( size(x), 1); a2 = zeros( size(x), 1); = 1:size(x)     a1(i) = x(i)*sin( 2*pi*(fc)*t(i));     a2(i) = x(i)*cos( 2*pi*(fc)*t(i)); end a1 = sqrt(a1); a2 = sqrt(a2); out = a1 + a2; 

i think modulation part right. problem think have don't have low-pass filter (lpf) demodulation. , should not calculate a1 , a2 directly. how add lpf demodulation code such output same original?

you need low-pass filter @ receiver after coherent demodulation, that's right. there's problem modulation. in example, symbol rate rs less angular carrier frequency w_c potentially causes overlapping of spectra @ receiver. consequently, reconstruction of information signal impossible. additionaly, in example fc * t = 2. means argument of sine function integer multiple of 2pi , therefore zero.

what need impulse shaper (can implemented low-pass filter) @ transmitter bandwidth w_g >= r/2. should so-called nyquist lowpass. carrier frequency must satisfy w_c > w_g.

i've written matlab script impulse shaping, modulation, demodulation, filtering , sampling, transmitted signal can reconstructed.

first define parameters, create random bits , mapping you've done. simple impulse response impulse shaping used, namely rectangular impulse. in real world we're going digital analog domain here, computer model, represent analog signal discrete 1 sampling frequency f_s. impulse shaper simple, because repeats each sample l times.

m = 16; % qam order fs = 16000; % sampling frequency in hz ts = 1/fs; % sampling interval in s fc = 1000; % carrier frequency in hz (must < fs/2 , > fg) rs = 100; % symbol rate ns = 20; % number of symbols  x = randint(ns, 1, m); y = modulate(modem.qammod(m), x);  l = fs / rs; % oversampling factor  % impulse shaping y_a = reshape(repmat(y', l, 1), 1, length(y)*l); 

now modulation. used carrier frequency satisfies above conditions: it's higher signal bandwidth , can still represented sampling frequency used.

%% modulation = real(y_a); q = imag(y_a); t = 0 : ts : (length(y_a) - 1) * ts; c1 = .* sin(2*pi * fc * t); c2 = q .* cos(2*pi * fc * t); s = c1 + c2; 

demodlation straightforward...

%% demodulation r_i = s .* sin(2*pi * fc * t); r_q = s .* -cos(2*pi * fc * t); 

to remove spectral tributaries @ 2f_c after demodulation low-pass filter required. used matlab fdatool create filter , part of following code. remember: signal bandwidth rs/2, , unwanted tributaries begin @ 2*fc - rs/2. how fpass , fstop found. (it might useful relax these requirements little bit.)

%% filter  % design filter least-squares method n     = 50;           % order fpass = rs/2;         % passband frequency fstop = 2*fc - rs/2;  % stopband frequency wpass = 1;            % passband weight wstop = 1;            % stopband weight  % calculate coefficients using firls function. b  = firls(n, [0 fpass fstop fs/2]/(fs/2), [1 1 0 0], [wpass wstop]);  % filtering w_i = filter(b, 1, r_i); w_q = filter(b, 1, r_q); 

after filtering still have sample received signal. here it's downsampling. used phase offset of l/2 avoid filter transitions.

%% sampling u_i = downsample(w_i, l, l/2); u_q = downsample(w_q, l, l/2); 

finally, plot constellation diagram , nice 16-qam constellation:

plot(u_i, u_q, '.'); 

you can find complete code here.

your question touched lot of topics of both dsp , matlab programming. not go detail everywhere. if have specific questions 16-qam modulation , demodulation place stack exchange site signal processing.