Hilbert Transforms in SSB Circuits
In simplest terms, a Hilbert Transform is any circuit that gives a 90 degree phase shift over a frequency range, with constant amplitude for all frequencies.
This is usually a phase difference network in which the two outputs are 90 o apart. Such a circuit, generated by elecfildes at www.elecfildes.com, is shown below for a frequency range of 50 - 5000 Hz, with a phase error of ±0.0607 o .
If
Vin = A * cos(ω t),
then
V1 = A * cos(ω t + θ)
and
V2 = A * cos(ω t + θ + 90 o)
In analyzing any circuit using a Hilbert transform, the trigonometric identities will be needed.
A1 cos( ωt) cos(ωt) = A1/2 *[ cos (ω- ω)t + cos (ω+ ω)t]
and
A1 cos (ωt) cos (ωt +90 ) = A1/2 * [cos (ωt - ωt -90) + cos (ωt + ωt +90)]
where ωt = 2*π*freq1*time and ωt= 2*π*freq2*time
Digital IIR and FIR Hilbert transformers can also be design with the IIR version using a bi-linear transform on the active circuit, and the FIR version would be directly designed by an exchange algorithm and is usually implemented in an FPGA.
Passive 90 degree phase splitter networks can be purchased as a module and usually have an octave of frequency range.
A passive topology would be generated, using the all pass transfer function
which in normalized in the example and assumes that C2 = L1 and a 1 Ω termination resistance.
For this first order all-pass lattice. Bartlett's Bisection Theorem is used to convert this lattice to a ladder as shown below.
The actual final implementation would be similar to
90 degree passive phase splitters are easily purchased as a module (see below).
The circuit below is a single sideband modulator. While a mixer normally produces sum and difference frequencies, a Hilbert transformer allows either the sum, or the difference, frequency to be selected.
If the Hilber transform produced only 89.9 degree shift at a certain frequency, then the phase error, θe, is 0.1 degree. The suppression of the unwanted sideband becomes
Suppression = -20*(Log )*tan(θe/2) dB.
The circuit below is an image rejecting mixer. While a mixer normally responds equally well to a frequency, and the image frequency, a Hilbert transformer allows either the frequency, or the image frequency, to be selected.
What is missing in this study is (1) References! and (2) Frequency dependence of the phase shift. Whithout such information the claim may be interesting but not fully convincing. Answer by mail to: vanco.litovski@elfak.ni.ac.rs, please.
My understanding (?) is that DSP does this much better than arrays of analog phase shift circuits. So finally SSB doesn't sound like a "slosh bucket" as the legendary Ashtabula Bill used to say.