Post by tmcdavid on Jan 12, 2007 11:24:22 GMT -5
I need to retrieve the base-band from an angle-modulated carrier.
- Background:
I recently read with interest the tutorial from this site that included the Hilbert Transform and the complex envelope concepts. Of particular note is the last paragraph that concludes “The Fourier transform of this representation will lead to the signal translated back down the baseband (and doubled with no negative frequency components) making it possible to get around the Nyquist sampling requirement and reduce computational load.”
I also studied the Discrete Hilbert Transform chapter of Oppenheim and Schafer (Digital Signal Processing, 1975) – unfortunately without full comprehension yet. But I noted the tight coupling of causality and the minimum-phase assumption to the validity of the Hilbert transform process. I was also interested by the phase-splitting approach mentioned near the end of the chapter, whereby a pair of filters, fed by the modulated data stream, have identical magnitude characteristics, but phases that differ by 90 degrees, so the two outputs constitute a Hilbert pair.
- The Intent:
I would like to use analog radio circuits to convert the received signal to an intermediate frequency (IF) of satisfactory amplitude for sampling and to use digital processes from there. I prefer to do that by under-sampling the band-limited IF with an appropriate A/D converter and processing the digital samples in a processor having efficient multiply-accumulate hardware for FIR filtering. The sampling rate would be such as to be more than adequate for the base-band spectrum, and would have the aliases of the sampling outside the bandpass of IF filter preceding the A/D. I desire to do the processing with FIR filters, versus taking in all the input samples and performing Fourier Transform processing.
- The Questions:
1) Can I sample the IF at a rate that will alias it all the way down to base-band (versus aliasing it to a digital IF that is centered at a frequency higher than the greatest instantaneous frequency deviation due to the modulation)? That question is intimately coupled to the next, which is:
2) Can the phase-splitting filters be low-pass (versus being forced to be band-pass by the discontinuity at DC that is bound to the Hilbert process)?
3) Will lack of causality in the modulation that creates the signal for demodulation be a problem? The modulation is presently achieved by convolving a zero-phase FIR symbol shape with the bit stream, so it is not strictly causal. Each bit has its energy spread over time equally on both sides of the bit center.
4) Will someone please direct me to the appropriate design guidelines and tools to help me construct a pair of phase-splitting FIR filters? My filter design experience is all zero-phase or linear-phase.
Regards, Terry
- Background:
I recently read with interest the tutorial from this site that included the Hilbert Transform and the complex envelope concepts. Of particular note is the last paragraph that concludes “The Fourier transform of this representation will lead to the signal translated back down the baseband (and doubled with no negative frequency components) making it possible to get around the Nyquist sampling requirement and reduce computational load.”
I also studied the Discrete Hilbert Transform chapter of Oppenheim and Schafer (Digital Signal Processing, 1975) – unfortunately without full comprehension yet. But I noted the tight coupling of causality and the minimum-phase assumption to the validity of the Hilbert transform process. I was also interested by the phase-splitting approach mentioned near the end of the chapter, whereby a pair of filters, fed by the modulated data stream, have identical magnitude characteristics, but phases that differ by 90 degrees, so the two outputs constitute a Hilbert pair.
- The Intent:
I would like to use analog radio circuits to convert the received signal to an intermediate frequency (IF) of satisfactory amplitude for sampling and to use digital processes from there. I prefer to do that by under-sampling the band-limited IF with an appropriate A/D converter and processing the digital samples in a processor having efficient multiply-accumulate hardware for FIR filtering. The sampling rate would be such as to be more than adequate for the base-band spectrum, and would have the aliases of the sampling outside the bandpass of IF filter preceding the A/D. I desire to do the processing with FIR filters, versus taking in all the input samples and performing Fourier Transform processing.
- The Questions:
1) Can I sample the IF at a rate that will alias it all the way down to base-band (versus aliasing it to a digital IF that is centered at a frequency higher than the greatest instantaneous frequency deviation due to the modulation)? That question is intimately coupled to the next, which is:
2) Can the phase-splitting filters be low-pass (versus being forced to be band-pass by the discontinuity at DC that is bound to the Hilbert process)?
3) Will lack of causality in the modulation that creates the signal for demodulation be a problem? The modulation is presently achieved by convolving a zero-phase FIR symbol shape with the bit stream, so it is not strictly causal. Each bit has its energy spread over time equally on both sides of the bit center.
4) Will someone please direct me to the appropriate design guidelines and tools to help me construct a pair of phase-splitting FIR filters? My filter design experience is all zero-phase or linear-phase.
Regards, Terry