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Lets have a RF-chain as above with system bandwith from HP corner to the green line say 1 MHz.

The signal accumulates delay as it passes through this analog chain, and due to it being non-liner filter chain, the group delay is likely as shown. Now at the end of LP1 and LP2 we have some form of detectors, which taken in the signal and do some type of functional detection on it. However the most important part of the detection is the critical control of switches s1 and s2 to determine when the signal is actually valid ( coming out of the two LFs ?).

So, the question would be what is a valid signal out of the LPF1 , since the input can not be a fixed single freq. component, otherwise we could just use the phase delay. Here the bandwidth is large upto 1 Mhz. So essentially we have to determine the optimum read-out freq or some statistical values to find the optimum switching times.

Can you please suggest how can one actually go on to do it ?
Without any new hardware design . Just lets try signal processing and statistical analysis


  • $\begingroup$ Are you transmitting a known signal? What are the characteristics of that signal? Optimum detection in that case would be to equalize the filter responses and correlate on the known signal assuming the signal has good autocorrelation properties. At the other extreme if your signal on versus off has a significant difference in power, then bandlimited power detectors could be used, each offset for the expected loss from each filter with a threshold between signal present and signal absent with the classic trade of probability of false alarm versus probability of detection. $\endgroup$ May 22 at 2:13
  • $\begingroup$ @DanBoschen Thanks for the response. Yes, Its a FMCW radar signal, so the the parameters would be known, except the beat freq(return time) of course. There is no hardware detector to be implemented, all has to be done in some form of signal statistics . Problem is different variations of radiated signal (slope, BW, chirp time) will be fed out (unknow to me, but i know the limits to them) & i need to find some optimum statistical way to enable the detectors without an actual priori knowledge of the signal in absolute terms, the only thing i know, is that its FMCW signal of certain BW & slope. $\endgroup$
    – BandW
    May 25 at 16:08
  • $\begingroup$ Ok I see- are you trying to compensate for hardware in your receiver that all signals will pass through or are you describing the effects of the different channels to each antenna which is the actual information of interest—- meaning are you trying to compensate these group delay variations away and they can be considered to be stationary or are you actively trying to measure and characterize the channels? $\endgroup$ May 25 at 16:50
  • $\begingroup$ @DanBoschen, well you put that nicely in context. Yes, I am trying to compensate only for the hardware in my receiver ( rest of the channels will be duplicated, or lets say i have only one channel for this example). The green line on the plots are the passband limits for the entire receiver and each sub-blocks. $\endgroup$
    – BandW
    May 26 at 6:58

The OP clarified in comments under the question that the intention is to compensate for group delay distortion introduced in the hardware. The typical approach to optimally compensate for this (in a least squared sense) with processing alone is to use the Wiener-Hopf equations to determine the coefficients of an equalizer that can be implemented with an FIR filter (meaning a difference equation with only feed-forward terms). I detail the full approach of doing this in these posts so will provide the links below, but to bottom line the process, the channel to be equalized (the receiver) is "sounded" with a known waveform that is spectrally rich (pseudo-random noise or frequency chirps are great choices as they also offer high average power or SNR; an impulse is a poor choice since it is a challenge to do that with high SNR), and with that and the received signal after the channel the reverse deconvolution can be computed in a least squared sense to determine the effective inverse channel, but importantly applicable to mixed phase systems (systems with both leading and trailing echoes) which on their own can't be inverted due to having zeros in the right-half plane. This approach would be ideally suited for distortions introduced in hardware implementations that are not changing with time (within an acceptable tolerance) since the channel can be sounded once and fixed coefficients determined relatively easily with pre-processing that can then be used without further modification, in contrast to the iterative algorithms that are needed to provide an adaptive solution for time-changing channels.

Details of the Wiener-Hopf Equations and shows application to determine the transfer function of the channel:

Compensating Loudspeaker frequency response in an audio signal

Equalizer Implementation Example by swapping Tx and Rx in the previous case, we can instead solve for the causal equalizer for a mixed phase system instead of the channel itself:

How determine the delay in my signal practically

  • $\begingroup$ Thankyou for your response. Your answer would be correct and very useful if i was in search for an channel/sub-block equalizer. Unfortonutely i think i wasnt careful in my last comment, I would like to elaborate . Lets say the BW is 5 MHz, and the received signal can be anywhere in the band, the switch s1 to be opened will depend on the tone(f1) received, the group delay(= HPF+LPF1)@f1 which is non-linear, similarly for switch s2 with groupdelay(=HPF+LPF1+LPF2)@f(1).... The equvilizer would be a good solution if i could place it before the switches and then have a linear values $\endgroup$
    – BandW
    May 26 at 13:11
  • $\begingroup$ @BandW I don’t quite understand; why can’t you equalize after the switch? Are you saying that even without discussing group delay variation, your components are non-linear? A linear system can have non-linear phase - it’s still a linear system in that we can swap the order of operations $\endgroup$ May 26 at 15:10
  • $\begingroup$ To be clear- the role and purpose of the equalizer is to compensate out the non-linear phase (which is the group delay variation you see) for that we can do the compensation on the final waveform as received and distorted— is there some other reason such a compensation would not be possible for you? $\endgroup$ May 26 at 20:19
  • $\begingroup$ yes, all your points are valid, but my concern is i can not equalize, don't have the HW or DSP change possible. Lets see refrence plane is at the input of HPF ( t=0), now the tone say 1 khz is delayed by 1 sec by hpf and then 3 sec by lpf1, 4 sec by lpf2, then i can say to switch s1, to close only at st=4, and st2=´8 sec, but for tone at 2 khz,, maybe st1=5sec, and st2=10sec. Since i dont know what tone is going to hit the HPF(but within system BW) , i dont know when to switch st1 and st2 on. $\endgroup$
    – BandW
    May 27 at 12:26
  • $\begingroup$ Now lets say, the tone is at 10 khz, the delay could be st1=20sec, and st2=30 sec, i dont want to swithc st1 on before 20sec for this tone, because the data going into detector will be not the actual signal, but if i wait 20 sec for st1 yes i will get the 10khz tone detected propertly, but because of waiting 20 sec, if there was a tone at 1 khz (delay st1=4sec) i would miss this tone. I want to know what would be a good statistical way to determine (within some error margin) an optimum switching timing for st1 and st2 for my defined BW. $\endgroup$
    – BandW
    May 27 at 12:28

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