# Delay estimation

Started by September 21, 2010
```Hello All,

There is a communication channel with nonminimum phase response. The
channel could be described as H(s)=P(s)/Q(s). The channel is equalized
by a linear equalizer. The equalization is near perfect but there is a
delay lag in the (equalizer + channel) due to the nonminimum phase part
of the response. What would be an efficient way to estimate this delay
lag, knowing the equalizer transfer function ?

DSP and Mixed Signal Design Consultant
http://www.abvolt.com

```
```Vladimir Vassilevsky  <nospam@nowhere.com> wrote:

>Hello All,

>There is a communication channel with nonminimum phase response. The
>channel could be described as H(s)=P(s)/Q(s). The channel is equalized
>by a linear equalizer. The equalization is near perfect but there is a
>delay lag in the (equalizer + channel) due to the nonminimum phase part
>of the response. What would be an efficient way to estimate this delay
>lag, knowing the equalizer transfer function ?

I like estimating the delay of a low-pass transfer function by
giving it a step input and measuring when it crosses the 50% point.
If it's a bandpass function I give it a windowed sinusoid and
measure the peak.

Or is this not what you're asking?

Steve
```
```
Steve Pope wrote:

>
>
>>Hello All,
>
>
>>There is a communication channel with nonminimum phase response. The
>>channel could be described as H(s)=P(s)/Q(s). The channel is equalized
>>by a linear equalizer. The equalization is near perfect but there is a
>>delay lag in the (equalizer + channel) due to the nonminimum phase part
>>of the response. What would be an efficient way to estimate this delay
>>lag, knowing the equalizer transfer function ?
>
>
> I like estimating the delay of a low-pass transfer function by
> giving it a step input and measuring when it crosses the 50% point.
> If it's a bandpass function I give it a windowed sinusoid and
> measure the peak.
>
> Or is this not what you're asking?

Leaving aside noise, fractional delays and minor technical detals.

The channel transfer function:

Hc(z) = Pc(z)/Qc(z)

If H(z) is minimum phase, then the equalizer transfer function:

He = 1/Hc(z) = Qc(z)/Pc(z)

So, Hc(z) He(z) === 1. Perfect equalization.

But, if the channel is nonminimal phase, the equalizer could only be
approximated by a stable function with finite amount of delay. Assumming
the approximation is close enough, we have:

Hc(z) He(z) ~ Z^-N

How to find N in efficient way if we know He(z) ?

DSP and Mixed Signal Design Consultant
http://www.abvolt.com

```
```On 9/21/2010 11:24 AM, Vladimir Vassilevsky wrote:
> Hello All,
>
> There is a communication channel with nonminimum phase response. The
> channel could be described as H(s)=P(s)/Q(s). The channel is equalized
> by a linear equalizer. The equalization is near perfect but there is a
> delay lag in the (equalizer + channel) due to the nonminimum phase part
> of the response. What would be an efficient way to estimate this delay
> lag, knowing the equalizer transfer function ?
>
> DSP and Mixed Signal Design Consultant
> http://www.abvolt.com
>
>
>

When you say that the equalization is near perfect that you mean the
amplitude part.  Is that right?  Or, does that description include phase?

I don't know if this makes sense and I'm sorry if it's a bit pedantic
but here goes:

Because you said that the channel has nonminimum phase response then it
could be linear phase and flat delay as one case of "nonminimum".  But I
don't imagine that helps much - just an observation to be clear.

And, to equalize the phase I imagine that you're trying to get to linear
phase / pure delay, right?

So, if the equalizer achieves this then the maximum delay of the
equalizer would match the minimum delay of the channel and vice versa -
plus a constant delay that may just come with the equalizer.

Channel delay: CL + CV where L means "linear phase" so it's a constant
and V means "variable" with frequency.

Equalizer delay: EL + EV which is known, right?

Total delay: CL + EL + CV + EV and we want CV + EV to be a constant as
well as CL + EL.

So, I would be tempted to find EL from the transfer function using a
staight line fit of some reasonable sort for the phase such that the
remaining part of the phase and, EV, can be calculated.  Then one could
estimate CV from:

I'm going to assume that EV is >0, then:

CV = K - EV where K is chosen such that CV >0.

This leaves the problem of the pure delay component of the channel and
maybe that's not too important....?  I don't know how that fits your
application needs.

Fred

```
```On Sep 21, 3:25&#2013266080;pm, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
> Steve Pope wrote:
> > Vladimir Vassilevsky &#2013266080;<nos...@nowhere.com> wrote:
>
> >>Hello All,
>
> >>There is a communication channel with nonminimum phase response. The
> >>channel could be described as H(s)=P(s)/Q(s). The channel is equalized
> >>by a linear equalizer. The equalization is near perfect but there is a
> >>delay lag in the (equalizer + channel) due to the nonminimum phase part
> >>of the response. What would be an efficient way to estimate this delay
> >>lag, knowing the equalizer transfer function ?
>
> > I like estimating the delay of a low-pass transfer function by
> > giving it a step input and measuring when it crosses the 50% point.
> > If it's a bandpass function I give it a windowed sinusoid and
> > measure the peak.
>
> > Or is this not what you're asking?
>
> Leaving aside noise, fractional delays and minor technical detals.
>
> The channel transfer function:
>
> Hc(z) = Pc(z)/Qc(z)
>
> If H(z) is minimum phase, then the equalizer transfer function:
>
> He = 1/Hc(z) = Qc(z)/Pc(z)
>
> So, Hc(z) He(z) === 1. Perfect equalization.
>
> But, if the channel is nonminimal phase, the equalizer could only be
> approximated by a stable function with finite amount of delay. Assumming
> the approximation is close enough, we have:
>
> Hc(z) He(z) ~ Z^-N
>
> How to find N in efficient way if we know He(z) ?
>
> DSP and Mixed Signal Design Consultanthttp://www.abvolt.com

Use a PN sequence of known good correlation properties (correlation
decaying fast for non-zero delays). Transmit this sequence as an input
signal through the composite channel (original channel + equalizer).
Cross-correlate the resultant output with the transmitted input and
pick the delay value that maximizes the cross-correlation.

Regards,
Dilip.
```
```>Hc(z) He(z) ~ Z^-N
>
>How to find N in efficient way if we know He(z) ?

N = center_of_gravity(1/He(z))

efficient way - direct channel impulse responce estimation via sounding of
the PN sequence with perfect PACF(Golay equences, Milewski equences)
```
```
Fred Marshall wrote:

> On 9/21/2010 11:24 AM, Vladimir Vassilevsky wrote:
>
>> Hello All,
>>
>> There is a communication channel with nonminimum phase response. The
>> channel could be described as H(s)=P(s)/Q(s). The channel is equalized
>> by a linear equalizer. The equalization is near perfect but there is a
>> delay lag in the (equalizer + channel) due to the nonminimum phase part
>> of the response. What would be an efficient way to estimate this delay
>> lag, knowing the equalizer transfer function ?
>>
> When you say that the equalization is near perfect that you mean the
> amplitude part.  Is that right?  Or, does that description include phase?

Amplitude and phase.

> I don't know if this makes sense and I'm sorry if it's a bit pedantic
> but here goes:
>
> Because you said that the channel has nonminimum phase response then it
> could be linear phase and flat delay as one case of "nonminimum".  But I
> don't imagine that helps much - just an observation to be clear.
>
> And, to equalize the phase I imagine that you're trying to get to linear
> phase / pure delay, right?
>
> So, if the equalizer achieves this then the maximum delay of the
> equalizer would match the minimum delay of the channel and vice versa -
> plus a constant delay that may just come with the equalizer.

common flat delay, so we leave it out of consideration.
However, some of the delay spread is due to minimum phase, and the other
is nonminimum. The questions are what would be the net delay after the
equalizer and if this delay could be derived from the equalizer function.

> Channel delay: CL + CV where L means "linear phase" so it's a constant
> and V means "variable" with frequency.
>
> Equalizer delay: EL + EV which is known, right?
>
> Total delay: CL + EL + CV + EV and we want CV + EV to be a constant as
> well as CL + EL.
>
> So, I would be tempted to find EL from the transfer function using a
> staight line fit of some reasonable sort for the phase such that the
> remaining part of the phase and, EV, can be calculated.  Then one could
> estimate CV from:
>
> I'm going to assume that EV is >0, then:
>
> CV = K - EV where K is chosen such that CV >0.
>
> This leaves the problem of the pure delay component of the channel and
> maybe that's not too important....?  I don't know how that fits your
> application needs.

So far I am estimating both transfer functions of the channel and the
equalizer, and deriving the delay from the combination of both. This is
neither elegant nor very efficient.