How to modulate a signal to a certain frequency in time domain?

Hello,

I am reading a paper where a root-raised cosine pulse signal of bandwidth 500 MHz (double-sided) is modulated at the frequency 6 GHz, their result looks like this :

If I understand well, to modulate the root raised cosine, it needs to be "multiplied" by a signal like h(t)=cos(ωt) with ω=2πf and f= 6 GHz.

I am trying to implement this same method to modulate a Butterworth pulse of 500 MHz bandwidth at a 8 GHz frequency.

My problem is that, in Matlab, when I multiply my pulse by a cosine this is not the result I get at all. I get something like this :

Could someone please explain what I might be doing wrong here.

Here is my code :

%% Create Butterworth pulse

% 1. Create a 4-th order Butterworth filter with 3-db bandwidth (cutoff) at 500 MHz
N = 4; % Filter Order
Fc = 500e6; % Cut off frequency is 500 MHz and it's a low pass butterworth filter
Fs = Fc*4; % 
[b,a] = butter(N, Fc/Fs); % Fc/Fs is the normalized cut off frequency

% 2. Pass an impulse to the Butterworth filter to create a Butterworth pulse

Tp = 2; % ns 
NTp = 8; % 
xSpan = NTp*Tp; % 
impulse = [1; zeros((NTp*4)-1), 1)]; % creating the impulse 
pulse = filter(b, a, impulse); % passing the impulse signal to the filter

% account for filter delay, center waveform at t=0 : by searching for the
% maximum and moving it to t=0
[~, idx] = max(pulse); % dummy ~ we ignore the "maximum value of the pulse vector" we just want its index idx 
len = length(pulse);
pulseCentered = [zeros(round(len/2)-idx, 1); pulse(1:(end-round(len/2)+idx))];

%% LO (Local Oscillator) signal cos(wt) at 7.9 GHz
f=7.9*10^(9);
A=1; % amplitude
w=2*pi*f;
t2 = (-xSpan/2:((xSpan/(len-1))):xSpan/2);
h=A.*cos(w*t2); % my cosine function
plot(t2,h);
xlabel('Time, (ns)');
ylabel('Local oscillator signal cos(wt), V');
title('LO signal f = 7.9 GHz')
grid on
axis tight

%% Multiplying my pulse response with the cosine function
m=(pulseCentered.*h);
figure
plot(m)
grid on
xlabel('Time, (ns)')
title('Modulated signal')

Thank you in advance for your help!

[Mathieu NOE]

you need to share your code if you really expect someone's help

[myoui]

Thank you for your comment. I have edited my post and added my code!

[Anna]

Mathematically speaking, modulation i.e. frequency shift is realized by multiplication by an exponent function in complex numbers:

M_{w} f(x)= e^{2 pi i w x} f(x)

It happens that the real part of a complex number is described by cos (while the imaginary part by sine)

due to the formula:

e^{ i w } = cos w + i sin w

hence you used only half of information.

[myoui]

Thank you for enlightening me about this! I will do this multiplication and report back!

[Anna]

what is "spc" in the line

impulse = [1; zeros((NTp*spc-1), 1)]; % creating the impulse

[myoui]

I have multiplied now by the exponential. I have plotted the abs of the resulting function as this function is also going to be complex. And here is the result below. It seems much better than what I had! About the enveloppe singal I have posted in my initial post, how to get that signal in my case?

Thank you!

m=(pulseCentered.*exp(i*2*pi*f*t2));
figure
plot(abs(m))
grid on
xlabel('Time, (ns)')
title('Modulated signal 7.9 GHz')

[myoui]

spc's value is 4, sorry I forgot to correct it.

I'm still not sure what spc is, I think maybe the symbol period. I have obtained the code for the Butterworth filter from Matlab example on Waveform Generation for UWB.