question on FIR channel

hi, suppose I have the FIR channel with 10 taps.
My target sampling rate is, for example, 50MHz, i.e., the sampling
period is 20ns.
But my channel taps do not fall on integer units of the sampling
period, say they fall on
33, 44, 55, 66, 77, 88, 99, 111, 122, 133 ns respectively.
What shall I do to create a FIR channel with taps falling exactly on
integer number of samplings? thank you in advance.

philgo wrote:
> hi, suppose I have the FIR channel with 10 taps.
> My target sampling rate is, for example, 50MHz, i.e., the sampling
> period is 20ns.
> But my channel taps do not fall on integer units of the sampling
> period, say they fall on
> 33, 44, 55, 66, 77, 88, 99, 111, 122, 133 ns respectively.

are you sure that is not 110, 121, and 132 ns?  everything else was a
multiple of 11 ns.

> What shall I do to create a FIR channel with taps falling exactly on
> integer number of samplings? thank you in advance.

so what you're trying to do is scale an existing FIR design from an FIR
filter that ran at a sampling rate of about 90.9 MHz or 1/(11 ns) to a
not-yet-existing FIR design that is meant to run at 50 MHz?  is that
it?

if so, it's a matter of interpolation and you might not end up with 10
taps.  you might have more, depending on the specs (how well you need
to nail some particular performance criteria).  if what you're
designing is an FIR with a given frequency response (that matches your
10-tap FIR at 90.9 MHz), then you should use a filter design program
(MATLAB and other numerical programs have this) to redesign the filter
with the same target response at the new sampling rate (and give up on
trying to adapt the other design to this).

r b-j

robert bristow-johnson wrote:
> philgo wrote:
> > hi, suppose I have the FIR channel with 10 taps.
> > My target sampling rate is, for example, 50MHz, i.e., the sampling
> > period is 20ns.
> > But my channel taps do not fall on integer units of the sampling
> > period, say they fall on
> > 33, 44, 55, 66, 77, 88, 99, 111, 122, 133 ns respectively.
>
> are you sure that is not 110, 121, and 132 ns?  everything else was a
> multiple of 11 ns.

I actually wanted to give some random numbers that are not integers of
20ns.

>
> > What shall I do to create a FIR channel with taps falling exactly on
> > integer number of samplings? thank you in advance.
>
> so what you're trying to do is scale an existing FIR design from an FIR
> filter that ran at a sampling rate of about 90.9 MHz or 1/(11 ns) to a
> not-yet-existing FIR design that is meant to run at 50 MHz?  is that
> it?

I do not get it. Basically, I am given an FIR channel at non-integer
samples and want to convert the FIR channel with only integer delayed
samples , but without changing the channel characteristics.

>
> if so, it's a matter of interpolation and you might not end up with 10
> taps.  you might have more, depending on the specs (how well you need
> to nail some particular performance criteria).  if what you're
> designing is an FIR with a given frequency response (that matches your
> 10-tap FIR at 90.9 MHz), then you should use a filter design program
> (MATLAB and other numerical programs have this) to redesign the filter
> with the same target response at the new sampling rate (and give up on
> trying to adapt the other design to this).

I agree it might not end up w ith 10 taps. Coud you be more specific on
how to generate the new channel from the old one? thanks

> 
> r b-j

philgo wrote:

   ...

> I agree it might not end up w ith 10 taps. Coud you be more specific on
> how to generate the new channel from the old one? thanks

Could be more specific about what you mean by "channel" in this context?

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

"philgo" <philgo@gmail.com> wrote in message 
news:1155700345.187970.125490@m79g2000cwm.googlegroups.com...
>
> robert bristow-johnson wrote:
>> philgo wrote:
>> > hi, suppose I have the FIR channel with 10 taps.
>> > My target sampling rate is, for example, 50MHz, i.e., the sampling
>> > period is 20ns.
>> > But my channel taps do not fall on integer units of the sampling
>> > period, say they fall on
>> > 33, 44, 55, 66, 77, 88, 99, 111, 122, 133 ns respectively.
>>
>> are you sure that is not 110, 121, and 132 ns?  everything else was a
>> multiple of 11 ns.
>
> I actually wanted to give some random numbers that are not integers of
> 20ns.
>
>>
>> > What shall I do to create a FIR channel with taps falling exactly on
>> > integer number of samplings? thank you in advance.
>>
>> so what you're trying to do is scale an existing FIR design from an FIR
>> filter that ran at a sampling rate of about 90.9 MHz or 1/(11 ns) to a
>> not-yet-existing FIR design that is meant to run at 50 MHz?  is that
>> it?
>
> I do not get it. Basically, I am given an FIR channel at non-integer
> samples and want to convert the FIR channel with only integer delayed
> samples , but without changing the channel characteristics.

I suppose you mean that you have either a real FIR filter that you are 
calling a "channel" or you have a multipath channel that is well 
approximated by a FIR filter with arbitrary delays between the paths.

Here are some suggestions for modeling such a "channel":

1) Carefully examine just how accurate the delays can be measured/specified. 
Then consider moving the delays within the range of error so that they fall 
on a regular grid - perhaps a sparsely populated regular grid.  That is, 
there can be lots of zero FIR coefficients.

2) Carefully consider sampling at a very high rate and then quantizing the 
times at which each of the nonzero FIR coefficients lie - such that the 
error introduced by quantizing delays is negligible.  Sort of the same thing 
as #1 but taken from slightly another perspective.

3) Consider randomizing the delays (or, if narrowband signals, the phase) to 
simulate channel behavior on a statistical basis.  This may be the most 
realistic simulation of all if the delays really aren't known that well or 
if they vary due to relative motion or a time-varying channel.

If the signal is narrowband then the original objective is may not be 
completely achievable.
If the signal is broadband then the original objective may be reasonably 
achievable.

Fred

philgo wrote:
> hi, suppose I have the FIR channel with 10 taps.
> My target sampling rate is, for example, 50MHz, i.e., the sampling
> period is 20ns.
> But my channel taps do not fall on integer units of the sampling
> period, say they fall on
> 33, 44, 55, 66, 77, 88, 99, 111, 122, 133 ns respectively.
> What shall I do to create a FIR channel with taps falling exactly on
> integer number of samplings? thank you in advance.

Let's say the input signal was x[n], and you wanted to compute

y[n] = sum_k a_k delay( x[n], t_k),

where the function delay(x[n], t) simply delays the signal x[n] by the
real value t, the t_k are the tap delays and the a_k are the tap
weights. The only problem is to compute the delay function. You can do
this by seperating t into integer and fractional part, ie.

t = m + f,

for example 3.14142... = 3 + sqrt(2) / 10. This way you can reduce the
delay function to compute delays that lie in [0, 1[ by using

delay(x[n], t) = delay(x[n-m], f).

That last function is called a fractional delay and can be implemented
using an FIR filter (with integer tap delays). An easy method to design
the filter is via windowed sinc. Do you need more help with this?

Regards,
Andor

robert bristow-johnson wrote:

> philgo wrote:

>>hi, suppose I have the FIR channel with 10 taps.
>>My target sampling rate is, for example, 50MHz, i.e., the sampling
>>period is 20ns.
>>But my channel taps do not fall on integer units of the sampling
>>period, say they fall on
>>33, 44, 55, 66, 77, 88, 99, 111, 122, 133 ns respectively.

> are you sure that is not 110, 121, and 132 ns?  everything else was a
> multiple of 11 ns.

At 90.49MHz they will round to those numbers.  That would
be an unusual clock, though.

-- glen

I wrote:
> philgo wrote:
> > hi, suppose I have the FIR channel with 10 taps.
> > My target sampling rate is, for example, 50MHz, i.e., the sampling
> > period is 20ns.
> > But my channel taps do not fall on integer units of the sampling
> > period, say they fall on
> > 33, 44, 55, 66, 77, 88, 99, 111, 122, 133 ns respectively.
> > What shall I do to create a FIR channel with taps falling exactly on
> > integer number of samplings? thank you in advance.
>
> Let's say the input signal was x[n], and you wanted to compute
>
> y[n] = sum_k a_k delay( x[n], t_k),
>
> where the function delay(x[n], t) simply delays the signal x[n] by the
> real value t, the t_k are the tap delays and the a_k are the tap
> weights. The only problem is to compute the delay function. You can do
> this by seperating t into integer and fractional part, ie.
>
> t = m + f,
>
> for example 3.14142... = 3 + sqrt(2) / 10. This way you can reduce the
> delay function to compute delays that lie in [0, 1[ by using
>
> delay(x[n], t) = delay(x[n-m], f).
>
> That last function is called a fractional delay and can be implemented
> using an FIR filter (with integer tap delays). An easy method to design
> the filter is via windowed sinc. Do you need more help with this?

On second thoughts, perhaps a more direct approach to model your filter
is with the windowing technique. If we assume the above impulse
response, the frequency response is given by

H(w) = sum_k a_k exp(-j w t_k).

You can sample H on a regular grid with M points, compute the inverse
DFT and then window with a window of size N<M to get the impulse
response (the regularly-tapped FIR filter coefficients). M, N and the
window are parameters that allow you to determine the grade of the
approximation.

> 
> Regards,
> Andor

Hi Andor,

Thanks for your reply.

Let me rephrase your reply first to see if I understand correctly.
Basically, the bottomline is to represent the fractional-delayed signal

delay(x[n-m], f) with 0<f<1

by a summation of integer-delayed signal

delay(x[n-m], k) across many different k's where k's are integers only.

To do so, we may reconstruct the signal delay(x[n-m], f] by a sinc
function delayed by time f. We then resample the delayed sinc function
on all integer positions and then get the coefficients to represent the
fractional-delayed signal by integer-delayed signal.

in mathematics, if we denote the signal as delta(x-f) as the impluse
with f being defined as above, then we are trying to reconstruct it
first as

sinc(x-f)=sin(x-f)/(x-f)

and then resample it to

..., sinc(-2-f)/(-2-f), sinc(-1-f)/(-1-f), sinc(0-f)/(0-f),
sinc(1-f)/(1-f), sinc(2-f)/(2-f), ...


therefore, to represent delta(x-f), we need to use

summuation {i from minus infinity to positive infinity, integers only}
sin(i-f)/(i-f) delta(x-i)

Is it right?

but am not very clear about it yet. Suppose we are trying to implement
it in Matlab,
could you please show some example?




Andor wrote:
> philgo wrote:
> > hi, suppose I have the FIR channel with 10 taps.
> > My target sampling rate is, for example, 50MHz, i.e., the sampling
> > period is 20ns.
> > But my channel taps do not fall on integer units of the sampling
> > period, say they fall on
> > 33, 44, 55, 66, 77, 88, 99, 111, 122, 133 ns respectively.
> > What shall I do to create a FIR channel with taps falling exactly on
> > integer number of samplings? thank you in advance.
>
> Let's say the input signal was x[n], and you wanted to compute
>
> y[n] = sum_k a_k delay( x[n], t_k),
>
> where the function delay(x[n], t) simply delays the signal x[n] by the
> real value t, the t_k are the tap delays and the a_k are the tap
> weights. The only problem is to compute the delay function. You can do
> this by seperating t into integer and fractional part, ie.
>
> t = m + f,
>
> for example 3.14142... = 3 + sqrt(2) / 10. This way you can reduce the
> delay function to compute delays that lie in [0, 1[ by using
>
> delay(x[n], t) = delay(x[n-m], f).
>
> That last function is called a fractional delay and can be implemented
> using an FIR filter (with integer tap delays). An easy method to design
> the filter is via windowed sinc. Do you need more help with this?
> 
> Regards,
> Andor

Hi Andor,

Thanks for your reply.

Let me rephrase your reply first to see if I understand correctly.
Basically, the bottomline is to represent the fractional-delayed signal

delay(x[n-m], f) with 0<f<1

by a summation of integer-delayed signal

delay(x[n-m], k) across many different k's where k's are integers only.

To do so, we may reconstruct the signal delay(x[n-m], f] by a sinc
function delayed by time f. We then resample the delayed sinc function
on all integer positions and then get the coefficients to represent the
fractional-delayed signal by integer-delayed signal.

in mathematics, if we denote the signal as delta(x-f) as the impluse
with f being defined as above, then we are trying to reconstruct it
first as

sinc(x-f)=sin(x-f)/(x-f)

and then resample it to

..., sin(-2-f)/(-2-f), sin(-1-f)/(-1-f), sin(0-f)/(0-f),
sin(1-f)/(1-f), sin(2-f)/(2-f), ...

therefore, to represent delta(x-f), we need to use

summuation {i from minus infinity to positive infinity, integers only}
sin(i-f)/(i-f) delta(x-i)

Is it right?

Andor wrote:
> philgo wrote:
> > hi, suppose I have the FIR channel with 10 taps.
> > My target sampling rate is, for example, 50MHz, i.e., the sampling
> > period is 20ns.
> > But my channel taps do not fall on integer units of the sampling
> > period, say they fall on
> > 33, 44, 55, 66, 77, 88, 99, 111, 122, 133 ns respectively.
> > What shall I do to create a FIR channel with taps falling exactly on
> > integer number of samplings? thank you in advance.
>
> Let's say the input signal was x[n], and you wanted to compute
>
> y[n] = sum_k a_k delay( x[n], t_k),
>
> where the function delay(x[n], t) simply delays the signal x[n] by the
> real value t, the t_k are the tap delays and the a_k are the tap
> weights. The only problem is to compute the delay function. You can do
> this by seperating t into integer and fractional part, ie.
>
> t = m + f,
>
> for example 3.14142... = 3 + sqrt(2) / 10. This way you can reduce the
> delay function to compute delays that lie in [0, 1[ by using
>
> delay(x[n], t) = delay(x[n-m], f).
>
> That last function is called a fractional delay and can be implemented
> using an FIR filter (with integer tap delays). An easy method to design
> the filter is via windowed sinc. Do you need more help with this?
> 
> Regards,
> Andor