The Modulated Wideband Converter

Conventional sub-Nyquist sampling methods for analog signals exploit prior information about the spectral support. A more challenging problem is spectrum-blind sub-Nyquist sampling of multiband signals. The modulated wideband converter (MWC) is the first system for sub-Nyquist sampling, which can be realized with existing devices and handle wideband analog signals.

Signal model. The Fourier transform of wideband signals often occupies only a small portion of a wide spectrum, with unknown frequency support. For example: in wideband communication, the receiver sees the sum of several radio-frequency transmissions. Each signal is modulated around an unknown and different carrier frequency.

The system. With an efficient hardware implementation and low computational load on the supporting digital processing, the modulated wideband converter (MWC) can blindly sample multiband analog signals at a low sub-Nyquist rate. The MWC first multiplies the analog signal by a bank of periodic waveforms. Then the product is lowpass filtered and sampled uniformly at a low rate. The waveform period and the uniform rate can be made as low as the expected width of each band, which is orders of magnitude smaller than the Nyquist rate. Reconstruction relies on recent ideas developed in the context of analog compressed sensing, and is comprised of a digital step which recovers the spectral support. The MWC enables baseband processing, namely generating a low rate sequence corresponding to any information band of interest from the given samples, without going through the high Nyquist rate. In the broader context of Nyquist sampling, the MWC scheme has the potential to break through the bandwidth barrier of state-of-the-art analog conversion technologies such as interleaved converters.

1. Matlab package of the analog system. The simulations of paper [4] were generated by this package.
    Usage: (A) unzip the package.
                (B) use start.m to run the simulations (can last many hours).
                (C) use GenFigs.m to generate the figures from the saved results.

2. A demo package of a digital counterpart system, which is easier for understanding and manipulation.
Usage: Unzip the package and run demo.m.

Overview

The modulated wideband converter (MWC) is comprised of two stages: sampling and reconstruction.

In the sampling stage, the input signal enters a bank of multiple channels simultaneously. In each channel, the input signal is multiplied by a mixing function. The mixing function is designed to be a periodic random bit-sequence. Its purpose is to spread the spectrum such that a portion of energy from each band appears in the baseband. After mixing, the signal spectrum is first truncated by a lowpass filter and then sampled at a low rate corresponding to the lowpass cutoff, yielding multiple channels of low-rate digital samples.

In the reconstruction stage, the low-rate digital samples enter the continuous-to-finite (CTF) block for spectrum support estimation. Once the unknown support of the spectrum is determined, the active bands of the input signal are reconstructed by digital processing and mature DAC technology.

Signal model

The signal of interest is a multiband signal consisting 3 pairs of bands (total N = 6 bands, as depicted in Fig. 1). Each band is of width B = 50MHz and of energy 1, 2, and 3. Ttime offsets are specifed by Taui, to better visualize the results. The Nyquist rate of the multiband signal is supposed to be as high as 10GHz.

SNR = 10;                                   % Input SNR
N = 6;                                      % Number of bands (when counting  a band and its conjugate version separately)
B = 50e6;                                   % Maximal width of each band
fnyq = 10e9;                                % Nyquist rate
Ei = [1 2 3];                               % Energy of the i'th band
Tnyq = 1/fnyq;
R = 1;                                      % The length of the signal is R*(K+K0)*L
K = 91;
K0 = 10;                                    % R*K0*L is reserved for padding zeros
L = 195;
TimeResolution = Tnyq/R;
TimeWin = [0  L*R*K-1 L*R*(K+K0)-1]*TimeResolution; % Time interval in which signal is observed
Taui = [0.7 0.4 0.3]*max(TimeWin);          % Time offest of the i'th band

fprintf(1,'---------------------------------------------------------------------------------------------\n');
fprintf(1,'Signal model\n');
fprintf(1,'   N= %d, B=%3.2f MHz, fnyq = %3.2f GHz\n', N, B/1e6, fnyq/1e9);

Sampling parameters

We use 50 low-rate sampling channels to sample the 10GHz multiband signal. The sampling rate for each channel is fs = 10GHz/195 = 51.3MHz. Therefore, the actual sampling rate is 51.3MHz*50 = 2.56GHz.

ChannelNum = 50;                            % Number of channels
L = 195;                                    % Aliasing rate
M = 195;
fp = fnyq/L;
fs = fp;                                    % Sampling rate at each channel, use fs=qfp, with odd q
m = ChannelNum;                             % Number of channels

% sign alternating  mixing
SignPatterns = randsrc(m,M);                % Draw a random +-1 for mixing sequences

% calucations
Tp = 1/fp;
Ts = 1/fs;
L0 = floor(M/2);
L = 2*L0+1;

fprintf(1,'---------------------------------------------------------------------------------------------\n');
fprintf(1,'Sampling parameters\n');
fprintf(1,'   fp = %3.2f MHz, m=%d, M=%d\n',fp/1e6,m,M);
fprintf(1,'   L0 = %d, L=%d, Tp=%3.2f uSec, Ts=%3.2f uSec\n', L0, L, Tp/1e-6, Ts/1e-6);

Signal Representation

Now we generate the input signal using the following codes. The carriers of the signal are randomly assigned.

t_axis = TimeWin(1)  : TimeResolution : TimeWin(end);     % Time axis
t_axis_sig  = TimeWin(1)  : TimeResolution : TimeWin(2);

fprintf(1,'---------------------------------------------------------------------------------------------\n');
fprintf(1,'Continuous representation\n');
fprintf(1,'   Time  window = [%3.2f , %3.2f) uSec\n',TimeWin(1)/1e-6, TimeWin(2)/1e-6 );
fprintf(1,'   Time resolution = %3.2f nSec, grid length = %d\n', TimeResolution/1e-9, length(t_axis));
fprintf(1,'---------------------------------------------------------------------------------------------\n');
fprintf(1,'Generating signal\n');
% Signal Generation
x = zeros(size(t_axis_sig));
fi = rand(1,N/2)*(fnyq/2-2*B) + B;      % Draw random carrier within [0, fnyq/2]
for n=1:(N/2)
    x = x+sqrt(Ei(n)) * sqrt(B)*sinc(B*(t_axis_sig-Taui(n))) .* cos(2*pi*fi(n)*(t_axis_sig-Taui(n)));
end
han_win = hann(length(x))';             % Add window
x = x.*han_win;
x = [x, zeros(1,R*K0*L)];               % Zero padding

% Calculate original support set
Sorig = [];
% explain:  we take the starting edges: fi-B/2  and divide by fp. minus
% 0.5 to shift half fp towards zero. then M0+1 is to move 0 = M0+1.
Starts = ceil((fi-B/2)/fp-0.5+L0+1);
Ends = ceil((fi+B/2)/fp-0.5+L0+1);
for i=1:(N/2)
    Sorig = union (Sorig,  Starts(i):Ends(i));
end
% Now add the negative frequencies
Sorig = union(Sorig, L+1-Sorig);
Sorig = sort(Sorig);

---------------------------------------------------------------------------------------------
Continuous representation
   Time  window = [0.00 , 1.77) uSec
   Time resolution = 0.10 nSec, grid length = 19695
---------------------------------------------------------------------------------------------
Generating signal

We run the following commands in MATLAB to see the spectrum of the signal.

>> figure, plot(linspace(-5,5,length(x)),abs(fftshift(fft(x)))),grid on 
>> title('Spectrum of x'), xlabel('Frequency (GHz)'), ylabel('Magnitude')

The indices of active bands are:

>> Sorig

Sorig=

11 12 58 59 64 65 131 132 137 138 184 185

Noise Generation

The white Gaussian noise is genereated such that its spectrum is in [-fnyq/2, fnyq/2].

noise_nyq = randn(1,(K+K0)*L);              % Generate white Gaussian nosie within [-fnyq/2,fnyq/2]
noise = interpft(noise_nyq, R*(K+K0)*L);    % Interpolate into a finer grid (the same length as the signal)

% Calculate energies
NoiseEnergy = norm(noise)^2;
SignalEnergy = norm(x)^2;
CurrentSNR = SignalEnergy/NoiseEnergy;

Mixing

Now we mix the input signal and noise with random periodical functions.

fprintf(1,'Mixing\n');

MixedSigSequences = zeros(m,length(t_axis));
for channel=1:m
    MixedSigSequences(channel,:) = MixSignal(x,t_axis,SignPatterns(channel,:),Tp);
end

MixedNoiseSequences = zeros(m,length(t_axis));
for channel=1:m
    MixedNoiseSequences(channel,:) = MixSignal(noise,t_axis,SignPatterns(channel,:),Tp);
end

The random mixing function aliases energies from each bands into the baseband. This is the key step that enables the following baseband operation without losing information.

>> figure,plot(linspace(-5,5,length(x)),abs(fftshift(fft(MixedSigSequences(1,:))))), grid on
>> title('Spectrum of a mixed signal'), xlabel('Frequency (GHz)'), ylabel('Magnitude')

Analog low-pass filtering and actual sampling

For each channel, we filter the mixed channel with an ideal low pass filter. Then we downsample the high-rate input sequences (19695 samples for each channel) into low-rate ones (101 samples for each channel).

fprintf(1,'Filtering and decimation (=sampling)\n');
% ideal pass filter
temp = zeros(1,K+K0);
temp(1) = 1;
lpf_z = interpft(temp,length(t_axis))/R/L; % impulse response of the low pass filter

SignalSampleSequences = zeros(m,K+K0);
NoiseSampleSequences = zeros(m,K+K0);
fprintf(1,'    Channel ');
decfactor = L*R;
for channel = 1:m
    fprintf(1,'.');  if ( (mod(channel,5)==0) || (channel==m)) fprintf(1,'%d',channel); end
    SignalSequence =  MixedSigSequences(channel,:);
    NoiseSequence   =  MixedNoiseSequences(channel,:);
    DigitalSignalSamples(channel, :) = FilterDecimate(SignalSequence,decfactor,lpf_z);
    DigitalNoiseSamples(channel, :) = FilterDecimate(NoiseSequence,decfactor,lpf_z);
end
Digital_time_axis = downsample(t_axis,decfactor);
DigitalLength = length(Digital_time_axis);

fprintf(1,'\n---------------------------------------------------------------------------------------------\n');
fprintf(1,'Sampling summary\n');
fprintf(1,'   %d channels, each gives %d samples\n',m,DigitalLength);

Filtering and decimation (=sampling)
    Channel .....5.....10.....15.....20.....25.....30.....35.....40.....45.....50
---------------------------------------------------------------------------------------------
Sampling summary
   50 channels, each gives 101 samples

CTF block

Next, we use CTF block to recover the support of active bands. The following results indicate that we successfully recover the support.

fprintf(1,'---------------------------------------------------------------------------------------------\n');
fprintf(1,'Entering CTF block\n');

% define matrices for fs=fp
S = SignPatterns;
theta = exp(-j*2*pi/L);
F = theta.^([0:L-1]'*[-L0:L0]);
np = 1:L0;
nn = (-L0):1:-1;
% This is for digital input only. Note that when R -> infinity,
% D then coincides with that of the paper
dn = [   (1-theta.^nn)./(1-theta.^(nn/R))/(L*R)      1/L    (1-theta.^np)./(1-theta.^(np/R))/(L*R)];
D = diag(dn);
A = S*F*D;
A = conj(A);

SNR_val = 10^(SNR/10);          % not dB
% combine signal and noise
DigitalSamples = DigitalSignalSamples + DigitalNoiseSamples*sqrt(CurrentSNR/SNR_val);

% Frame construction
Q = DigitalSamples* DigitalSamples';
% decompose Q to find frame V
NumDomEigVals= FindNonZeroValues(eig(Q),5e-8);
[V,d] = eig_r(Q,min(NumDomEigVals,2*N));
v = V*diag(sqrt(d));
% N iterations at most, since we force symmetry in the support...
[u, RecSupp] = RunOMP_Unnormalized(v, A, N, 0, 0.01, true);
RecSuppSorted = sort(unique(RecSupp));
% Decide on success
if (is_contained(Sorig,RecSuppSorted)  && (rank(A(:,RecSuppSorted)) == length(RecSuppSorted)))
    Success= 1;
    fprintf('Successful support recovery\n');
else
    Success = 0;
    fprintf('Failed support recovery\n');
end

---------------------------------------------------------------------------------------------
Entering CTF block
Successful support recovery

Recover the signal

Once we determine the support, we can easily reconstruct each signal band by a direct pseudoinverse operation. The signal bands are then modulated to their corresponding carrier frequencies.
All the band signals are then modulated to their corresponding carrier frequencies.

A_S = A(:,RecSuppSorted);
hat_zn = pinv(A_S)*DigitalSamples;  % inverting A_S
hat_zt = zeros(size(hat_zn,1),length(x));
for ii = 1:size(hat_zt,1)                     % interpolate (by sinc)
    hat_zt(ii,:) = interpft(hat_zn(ii,:),L*R*length(hat_zn(ii,:)));
end

x_rec = zeros(1,length(x));
for ii = 1:size(hat_zt,1)                     % modulate each band to their corresponding carriers
    x_rec = x_rec+hat_zt(ii,:).*exp(j*2*pi*(RecSuppSorted(ii)-L0-1)*fp.*t_axis);
end
x_rec = real(x_rec);					      % Reconstructed signal
sig = x + noise*sqrt(CurrentSNR/SNR_val);     % Original noised signal

Analysis & Plots

We finally plot all the results here.

figure,
plot(t_axis,x,'r'), axis([t_axis(1) t_axis(end) 1.1*min(x) 1.1*max(x) ])
title('Original signal'), xlabel('t')
grid on
figure,plot(t_axis,sig,'r'), axis([t_axis(1) t_axis(end) 1.1*min(x) 1.1*max(x) ])
title('Original noised signal'), xlabel('t')
grid on
figure, plot(t_axis,x_rec), axis([t_axis(1) t_axis(end) 1.1*min(x) 1.1*max(x) ]),
grid on,
title('Reconstructed signal'), xlabel('t')

The output SNR is

 >> 20*log10(norm(x)/norm(x-x_rec))

ans = 
 14.9204

Therefore, compared to the conventional Nyquist sampling , the SNR gain is 14.9204 - 10 = 4.9204 (dB).

Note that the simulation above is for a noisy input signal. If the signal is noiseless (this can be implemented by assigning a sufficiently large value to the parameter SNR), then by re-doing the simulation, the pure reconstruction error is calculated to be about 1e-6, which is almost perfect reconstruction.