🚀 go-pugleaf

In article <TIM.93Jan10124435@ear-ache.mit.edu> tim@ear-ache.mit.edu (Tim  
Wilson) writes:
> 880840m@axe.acadiau.ca (MICHAEL ALEXANDER MCKAY) writes:
> 
>    Is there a time-domain filter which, given a real signal as input,  
will
>    output the corresponding imaginary signal?
>    I've tried to make a filter with unity gain and pi/2 phase shift for
>    all frequencies, but with no luck.
> 
> See Oppenheim and Shafer, _Digital Signal Processing_ (1st edition),
> Section 7.4:  "Hilbert Transform Relations for Complex Sequences."  In
> particular, see Section 7.4.1, "Design of Hiblert Transformers."
> I quote:  "...the ideal Hibert transformer or 90-degree phase shifter
> takes its place alongside the ideal lowpass filter and ideal
> bandlimited differentiator as valuable theoretical concepts which
> correspond to noncausal systems and for which the system function
> exists only in a restricted sense."
> 
> In the same section, O&S discuss finite-duration approximations of the
> ideal Hilbert transformer using frequency-sampling, windowing, and
> equiripple approximations.  Finally, they reference B. Gold, A. V.
> Oppenheim, and C. M. Rader, "Theory and Implementation of the Discrete
> Hibert Transform," _Proc. Symp. Computer Processing in
> Communications_, Vol. 19, Polytechnic Press, 1970, New York for a
> recursive implementation of a 90-degree phase-splitting system.
> 
> Good luck.
> --
> Tim Wilson
> Internet:  tim@ear-ache.mit.edu
> UUCP:  mit-eddie!mit-athena!tim

The Hilbert filter approach is the "correct" way to derive an analytic  
equivalent to a real-valued signal. However, if you can assume that your  
PASSBAND signal is also narrowband, there is an alternative. Assume that  
we have a passband centered at a frequency fc having a bandwidth BW and  
that the signal has been sampled at a frequency fS. Furthermore, the  
narrowband assumption implies that fc >> BW. Then, given the data stream  
s(k), we would:

1) Rotate the upper sideband to DC via 

                 y(k) = s(k)*exp(-j 2 pi fc k/fS)

2) Apply a low-pass filter to chop out the lower sideband (which is now  
centered on -2fc) and you have an basebanded analytic equivalent of your  
input signal. Make that low-pass filter of the decimating FIR type and you  
are probably picking up significant computational efficiencies.

This techniques is approximately a digital equivalent of the quadrature  
sampling approach and is valid whenever quadrature sampling would be  
appropriate. (However, these days a digital implementation takes up less  
space than the analog so for embedded systems, the digital is generally  
preferable.)

Hope this is of some use,

Mark.
--
Mark Kotanchek
Guidance & Control Dept - 363 ASB
Applied Research Lab/Penn State
P.O. Box 30
State College, PA 16804