How to compute a complex modulated signal from an audio file?

Question

I have a 30-seconds wav audio file with a sample rate of 44100Hz, obviously this array of samples is a 1D array, and so when I modulate it in AM (Amplitude Modulation) I get back a 1D array, but I want it to return a 2D array of complex values, I don't really know how can you generate the real and imaginary part from one value and so I ended up here.

Here is my code:

SAMPLE_FOR = 30 # in seconds
samplerate, data = scipy.io.wavfile.read(r'Recording.wav')
data = data[0:int(samplerate*SAMPLE_FOR)]
fm = generateSignalFM(data,time)

def generateSignalAM(slc,t):
    samples,time_vec = SamplerateConversion(slc)
    w = low_cut_filter(samples,176400,22050)

    TWO_PI = 2 * np.pi
    carrier_hz = 20000
    ac = 0.5

    carrier = np.sin(TWO_PI * carrier_hz * time_vec)
    envelope = (1.0 + ac * w)
    modulated = envelope * carrier
    return modulated

And FM signal code:

def generateSignalFM(samples,t, fc=None, b=.3):

    w = low_cut_filter(samples,fs,22050)

    N = len(w)
    if fc is None:
        fc = 75000 

    x0 = w[:N]
    # ensure it's [-.5, .5] so diff(phi) is b*[-pi, pi]
    x0 /= (2*np.abs(x0).max())

    # generate phase
    phi0 = 2*np.pi * fc * time_vec
    phi1 = 2*np.pi * b * np.cumsum(x0)
    phi = phi0 + phi1
    diffmax  = np.abs(np.diff(phi)).max()
    # `b` correction
    if diffmax >= np.pi:
        diffmax0 = np.abs(np.diff(phi0)).max()
        diffmax1 = np.abs(np.diff(phi1)).max()
        phi1 *= ((np.pi - diffmax0) / diffmax1)
        phi = phi0 + phi1

    # modulate
    x = np.cos(phi)
    return x

Answer 1

If the signal is a real audio waveform that only has AM components, the baseband equivalent waveform will be real unless the carrier has a phase offset. If the intention is to model the effect of carrier phase offset, this can be accomplished by multiplying the baseband waveform with $e^{j\phi}$ for a given carrier offset $\phi$ resulting in a complex signal.

If the intention is to make an analytic signal (positive half spectrum only), the relationship is given by:

$$X(\omega) \Leftrightarrow x(t) = I(t)+jQ(t)$$

Where $x(t)$ is the complex analytic signal, $I(t)$ is the original real waveform, and $Q(t)$ is the Hilbert Transform of $I(t)$. $X(\omega$) is the Fourier Transform of $x(t)$ and will have positive frequency components only.

The analytic waveform can be created directly with the OP's code by changing the carrier from a real carrier as given by

carrier = np.sin(TWO_PI * carrier_hz * time_vec)

To a complex analytic signal carrier as given by:

carrier = np.exp(1j * TWO_PI * carrier_hz * time_vec)

This is because $e^{j\omega t}$ is the analytic signal for $\cos(\omega t)$. Compare the following given by Euler's equation to the general relationship previously given above for the real signal $I(t)$ and its analytic signal $x(t)$:

$$ e^{j\omega t} = \cos(\omega t) + j\sin(\omega t)$$

Here, $e^{j\omega t}$ is the analytic signal, $\cos(\omega t)$ is the original real carrier, and $\sin(\omega t)$ is the Hilbert transform of $\cos(\omega t)$.