Standard methods of explaining modulation seem to fall flat on their faces when we start to talk about things like singlesideband, in fact, when we talk about sidebands at all. A much more natural way to understand modulation is to describe it at the baseband level using a unit circle. At this level, a single frequency is represented by a rotating vector of constant speed (angular velocity). An unmoving vector is simply a DC offset, thus, a carrier wave.
One of the traps here is that this does involve complex numbers, and therefore, imaginary amplitudes and negative frequencies. Regardless, I believe that this is the easiest way to understand and synthesize modulated waveforms for transmission in zero-IF systems which include most SDRs (Software Defined Radios).
What do these examples do?
These examples are designed to be played back either directly in an SDR application (as a RAW IQ file) or transmitted using, for example, a hackRF using hackRF_transfer.exe. This allows us to sanity check our waveforms and even see things like bandwidth, carrier level, and filtering etc.
The most 'natural' wave to generate is "Double Side Band". This is basically AM (amplitude modulation) sans carrier wave. This means that the instantaneous output power is directly proportional to the input signal's instantaneous amplitude -- silence on the input means the transmitter is effectively turned off.
We will modulate the amplitude of I with the input signal by the modulation constant K
- K = Gain => Gain can be whatever it needs to be here, it will only affect the average output power.
- I = K*Signal => The real axis will be set to the input signal times a gain constant.
- Q = 0 => we will NOT modulate the imaginary axis at all.
- Because we set Q to 0, we will see both positive AND negative parts of all frequency components. (See Euler's equation for cos(x) in terms of e^jw)
- If we input pure sinusoids (like we will get from an audio signal), we will have NO CARRIER.
%% Generate DSB clear all; K = 1; %Modulation Constant Fs = 1e6; %Sample Rate. 1e6 is the minimum sample rate accepted by HackRF Ts = 1/Fs; %Sample Period t = [0:Ts:10]; %Time axis (0 to 10 seconds at Ts interval) %Generate our input signal, which MUST be sampled at (or resampled to) Fs %We have options! Constant frequency, sinusoidally fading in and out at a constant frequency, and frequency sweep %Feel free to play here! fs = 1000; %Signal frequency Signal = sin(2*pi*fs.*t); %fs = 1000; %Signal frequency %Signal = cos(2*pi*(1/10).*t) .* sin(2*pi*fs.*t); %This is our input signal, which MUST be sampled at Fs as presented here %fs = linspace(1e3,10e3,numel(t)); %Signal frequency, from 1khz to 10khz (linearly spaced array of increasing frequency values) %Signal = sin(2*pi*fs.*t); I = K*Signal+1; %Modulate I Q = 0; %Do not modulate Q to preserve symmetric upper and lower sideband (neg and pos frequencies) dataStream = I + Q*1i; %% Format data for playback on hackrf idxo=1; scale = max(real(dataStream)); for idx=1:numel(t) dataStreamFile(idxo) = int8(real(dataStream(idx))*(127/scale)); dataStreamFile(idxo+1) = int8(imag(dataStream(idx))*(127/scale)); idxo = idxo+2; end %% Write to RAW file fhandle = fopen('output.iq','w'); fwrite(fhandle,dataStreamFile,'int8'); fclose(fhandle);