Testing a Hilbert transform on a real signal to transmit it as Single SideBand

Question

I am testing some Hilbert transform theory in practice. On the practical side I use Python and a soundcard to generate baseband I and Q signals. And an IQ modulator to upconvert with 250kHz. Lastly a spectrum analyzer to inspect the result.

I was under the impression that if you take a real signal, calculate its Hilbert transform, multiply that transform by j and add that to the original signal that you can transmit that as SSB by simply upconverting it.

So I wrote this script:

import numpy as np
from scipy import signal
import sounddevice as sd

def tx(I,Q, d=duration):
    # 'play' I and Q on soundcard (stereo)
    assert len(I)==len(Q), f"inequal IQ lengths"
    assert len(I)>=fs*duration, f"IQ columnlength ({len(I)/fs}s) too short for duration ({duration}s)"
    samples = int(d*fs); I=I[:samples]; Q=Q[:samples]
    v = .1  # volume
    IQ = np.column_stack((v*I, v*Q))
    sd.play(IQ, fs)
    sd.wait()

fs = 32768
duration = 4

# create 5 tones
t = np.arange(int(fs*duration)) / fs
tones = [440,880,1200,1500,2000]
s = 0; 
for i, f in enumerate(tones):
    s += np.cos(2*np.pi*f*t)
s = s/np.max(s)   # reduce power to avoid clipping

# play original signal 
I = np.real(s); Q = np.imag(s); tx(I,Q, duration)

# play s + j times the hilbert transform of s
h = signal.hilbert(s)
y = s + 1j*h
I = np.real(y); Q = np.imag(y); tx(I,Q, duration)

# play just the hilbert transform of s
I = np.real(h); Q = np.imag(h); tx(I,Q, duration)

Transmission as said is via the audio left and right channel to the I and Q input of the 250kHz upconverter.

The three results of the code are:

s #original signal:

s + 1j*signal.hilbert(s):

signal.hilbert(s):

Clearly things are not as I expected for the last two scenarios. What am I doing wrong?

Answer 1

As commented by Matt L, scipy.signal.hilbert does not output the Hilbert transform itself, but the analytical signal (by using a Hilbert transform).

Same goes for Matlab.