How to generate random data with a specific bandwidth

Question

I generate some random data (bits, actually, using MATLAB's randi) and then I convert such bits to BPSK (1 and -1). After that, I use a RRC pulse (roll-off factor of 0.35, 8 symbols and 4 samples per symbol) to create the waveform.

The bandwidth of the pulse-shaped BPSK then would be ($\beta$ is the roll-off factor):

$$ BW = R_S(1+\beta) $$

My problem here is how to determine $R_s$ (data rate). I mean, I'm using randi, not an actual data source so initially I tried to make that value up. I suppose $R_s=1MHz$, after pulse shape the sampling frequency is $f_s = sps*R_S = 4MHz$ ($sps=4$ because of the 4 samples per symbol stated above), then the bandwidth should be something like $BW=(1+0.35)1MHz=1.35MHz$ and tried to plot the DFT of the pulse-shaped sequence for a final sampling rate of $4 MHz$:

As you can see, the bandwidth is nowhere near $1.35 MHz$.

So these are my questions:

1. How can I manipulate/control $R_S$
2. What's wrong with my calculations?

Thanks!

Answer 1

How can I manipulate/control $R_S$?

You usually start with a desired pulse rate $R_S$. Then, the number of samples per symbol is $f_sT_S$, where $f_s$ is the sampling rate and $T_S = 1/R_S$. The resulting signal bandwidth will be $B = (1+\beta)R_S/2$, where $\beta$ is the excess bandwidth in the pulse you choose to use.

In other cases you have a desired bandwidth $B$. In that case, the pulse rate is determined from $R_S = 2B/(1+\beta)$. The sampling rate is again determined by how many samples per symbol you want.

If you go with higher-order or quadrature modulation, then each pulse will transmit $k=2^M$ bits, with $M$ the number of pulse amplitudes allowed. Then, $R_S = R_b / k$, where $R_b$ is the bit rate. The other equations above remain the same.

What's wrong with my calculations?

You missed a factor of $1/2$ in the bandwidth formula. With a rate $R_S = 10^6$ bits per second, using sinc pulses ($β$=0), you'd need $B = 500\text{ kHz}$. With excess bandwidth $0.35$, you need $B = 500 \times 1.35 = 675\text{ kHz}$, which is consistent with your result.

For completeness, here's some Matlab code and the resulting plot:

b = randi([0,1],10000,1); % generate bits
a = 2*b-1;  % generate pulse amplitudes
Rs = 1; % define the pulse rate
Ts = 1/Rs;
fs = 4; % define the sampling frequency
beta = 0.35; % define the EBW
p = rcosdesign(beta, 20, Ts*fs); % raised cosine pulse with 4 sps and 0.35 EBW
% Now we upsample the amplitudes by a factor fs.
a_up = upsample(a,fs);
% The transmitted signal is the convolution of a_up with p
s = conv(p,a_up);
% Let's see the spectrum of s:
S = fft(s);
L = length(s);
P2 = abs(S/L);
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);
f = fs*(0:(L/2))/L;
plot(f,P1);
% Plot a line at the bandwidth
hold on; plot([0.675, 0.675], [0,0.02],'r'); hold off;

Answer 2

1. In the case of BPSK, the data rate equals the symbol rate (one bit per symbol). One way to control $R_S$ is to change the modulation. For example, you can increase the number of bits per symbol by choosing a higher order modulation like 16-QAM which has 4 bits per symbol.

2. There is nothing wrong with your calculation. The plot is a single sided spectrum so you only see half the bandwidth. It roughly dies off around half of 1.35 MHz so assuming your code is correct, I see nothing particularly wrong with the plot or calculation.