Hi!

I was trying to understand and derive the math for I/Q-demodulation of AM-SSB SC and ended up with this equation:

$$y(t) = H(x(t)) sin(\theta) + x(t) cos(\theta)$$

and I get the Fourier-transform 

$$Y(\omega) = e^{-sgn(w) \, j \, \theta} X(w)$$

H() is the hilbert-transform

With theta = pi/2, y(t) will be the hilbert-transform of x(t) and with the known property of generating a phase shift of pi/2. But with an arbitrary value of theta, y(t) seems to be a phase shifted signal of x(t) by theta.

I hope the theory and math is correct. However, I could not find much when searing in this topic. Does anyone knows, is this kind of filtering normally used in any application?

#SDR

/Fredrik

Here is the math and operation of the SSB demodulator (and yes such filtering with a Hilbert transform is commonly done).

The input signal as a real and therefore two-sided complex conjugate spectrum (consisting of both positive and negative frequencies in the passband of equal magnitude and opposite phase) is converted to a one-sided positive frequency only spectrum (referred to as the "analytic signal") using the Hilbert Transform and the relationship:

$$x_a(t) = x(t) + j \hat{x}(t)$$

Where \(x(t)\) is the real input signal, and \(\hat{x}(t)\) is the Hilbert Transform of \(x(t)\).  

To bottom-line the math for SSB, the operation is to take the real part of the product of the analytic signal given above with a complex local oscillator which shifts the desired upper-sideband signal from passband (just above the carrier) to baseband (just above DC):

$$Re\{x_a(t)e^{-j\omega_c t}\}$$ 

$$=Re\{(x(t) + j \hat{x}(t))(\cos(\omega_c t)- j\sin(\omega_c t))\}$$ 

$$= x(t)\cos(\omega_c t) + \hat{x}(t)\sin(\omega_c t)$$

Further details for the weary:

For SSB (using upper sideband for this description) , we are interested in down-converting the spectrum just above the carrier frequency to baseband. Let's consider our original signal before modulation as a simple low frequency sinusoid with radian frequency \(\omega_1\), that once modulated using SSB would reside just above the carrier at \(\omega_c\) with frequency \(\omega_c + \omega_1\) (assuming we use upper sideband). In this example \(x(t)\) would be:

$$x(t) = \cos((\omega_c + \omega_1)t)$$ 

This is a real sinusoid, and thus it's Fourier Transform consists of positive and negative frequency components, specifically an impulse at \(\omega_c + \omega_1\), and another at \(-\omega_c - \omega_1\). It may help to visualize the form \(e^{j\omega t}\) as a "spinning phasor" in the time domain on the complex IQ plane rotating in one direction counter-clockwise (like a bicycle wheel) and therefore represents a positive frequency given the phase versus time increases linearly at a positive rate. The Euler relationship \(2\cos(\omega t) = e^{j\omega t} + e^{-j\omega t}\) really ties this together well showing the positive and negative frequency components of a sinusoid.

We use the Hilbert Transform with the first formula I gave to produce two outputs (real and imaginary) as a single complex waveform given as:

$$x_a(t) = \cos((\omega_c + \omega_1)t) + j\sin((\omega_c+\omega_1)t)$$

The Hilbert Transform of a cosine is a sine, so when we use the Hilbert Transform to create the analytic signal, our real \(x(t)\) is converted to a complex \(x_a(t)\), which then requires two signal traces to build, commonly in rectangular form as \(I\) and \(Q\). In the analog world we can use a "Quadrature Splitter" from Mini-circuits and others to realize this resulting positive-only spectral representation of /(x(t)/) - a passband signal still residing at /(\omega_c + \omega_1/) but only in the positive frequency axis. Thus, using Euler's relationships again but seeing the full power of the "analytic signal" we see that \(x_a(t)\) is simply the positive only frequency component of \(x(t)\):

$$x_a(t) = \cos((\omega_c + \omega_1)t) + j\sin((\omega_c+\omega_1)t) = e^{j(\omega_c+\omega_1)t}$$

Highlighting this:

$$x_a(t) = e^{j(\omega_c+\omega_1)t}$$

When we build it, we use the expanded rectangular form, but given the products involved, the polar form shown makes it very intuitive for analysis and visualization. With that I often think of the two traces (I and Q) commonly used in implementation to represent one single complex waveform - greatly simplifying the math and intuition. 

With the analytic signal in hand, we can now apply the complex down-conversion from above to shift this passband signal to baseband.

Similarly, the local oscillator, which starts as a real \(\cos(\omega_c t)\), also uses the Hilbert to convert it to a "single spinning phasor", rotating as a negative frequency (conjugate of the analytic LO) which serves to translate the passband to baseband with no image filtering required:

Local Oscillator:

$$e^{-j\omega_c t} = \cos(\omega_c t) - j\sin(\omega_c t)$$

Complex Product for down-conversion:

$$x_a(t)e^{-j\omega_c t} = e^{j(\omega_c + \omega_1)t}e^{-j\omega_c t} = e^{j\omega_1 t}$$

Simple here given we just sum the exponents when we multiply the exponentials.

The result of this is a complex IQ signal at baseband representing the analytic signal equivalent of our original signal: \(e^{j\omega_1t}\). At this point, we can take the real component to recover \(\cos(\omega_1 t)\) with no aliasing or imaging issues. The complex product requires four real multipliers and two adders in implementation, but here since we only needed the real component, as first introduced, we only need to use two multipliers and one adder. If the original signal was a complex baseband signal such as QAM, we would use all four multipliers to maintain a complex baseband IQ signal.

Following that process, and especially the spectrums at each step, is very insightful.  I detail that further including the "reality" of complex signals here: https://dsp.stackexchange.com/a/91803/21048

Further insights with complex signals:

All complex numbers are represented as an ordered pair of real numbers, such as rectangular form \((I, Q)\) or polar form \((r, \phi)\), whether it be on paper in a mathematical analysis, or in real hardware where we build and use complex waveforms. We relate the two forms as:

$$re^{j\phi} = r \cos(\phi) + j r\sin(\phi) = I + j Q$$

With \(r = \sqrt{I^2 + Q^2}\) as the radius.

The "j" is simply a label to designate which of the two real numbers in the ordered pair represents the imaginary component of the resulting complex number. 

Hi!

Maybe I was a little short in my question. The equation

$$y(t) = H(x(t)) sin(\theta) + x(t) cos(\theta)$$

arises in the case there is a (constant) phase difference between the local oscillator at the receiver and the transmitter. 

x(t) is the 'original' message signal, theta the phase difference by the LO's and y(t) is the demodulated signal. 

My question was not necessary about SSB but rather about the equation itself that seems to produce an phase shift by theta. Is there any application where you use an arbitrary phase shift (not necessary pi/2) with this kind of equation?  

/Fredrik

Hi Fredrik-

If x(t)was a real passband signal, then the math as you show it could be used in a carrier synchronization approach to correct for phase and frequency offsets directly at passband. 

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Hello!

Ok, thanks!

I was also wondering about some physical interpretation of the phase shift due to that different frequencies will be delayed in time different. I was playing around in gnu-radio having a signal with 2 different frequencies. In the time plot you can see that the shape of signal changes with phase but sending it to sound card I cant tell if I could hear any difference.

Fredrik 

AI:

Human hearing is generally considered insensitive to phase when listening to a single sound source (monaural). Our ears are primarily sensitive to the frequency spectrum (loudness/pitch) rather than the phase alignment of harmonics. However, the brain is highly sensitive to phase differences between the two ears (binaural) for localizing sound direction at low frequencies.

Hi Fredrik-

With signal processing many conflate phase with time delay, given our first introduction to "phase" with two sine waves separated in time. While this relationship works at one frequency, it has huge pitfalls if that is what is in our head when we look at systems like this using the Hilbert transform and complex signal processing. I detail this at the beginning of this presentation on control loops, the first part is more generically to address this issue. Ultimately "phase" is a rotation on the complex plane, and every sample in a complex waveform has a magnitude and phase. That waveform can be samples representing any units (volts, amps, counts, squirrels...) versus time, as a complex time domain waveform, or samples versus frequency as a frequency domain waveform (or we can have other domains such as codes...).