Frequency domain interpolation Question

This problem comes from my simulation of Single Carrier Frequency Domain
Equalization(SC-FDE) system.

Suppose the symbol rate is Ts, for reason of pulse shaping filtering I
perform fractional space sampling with sample rate Ts/2. A FFT block length
is 512 and the pilot(used for channel estimation) length is 64. Channel
estimation is done as follows:

(1) use length 64*2(sampling rate Ts/2) pilot data to estimation channel
frequency response H;
(2) IFFT transforms H to time response h;
(3) padding h with zeros to length 256*2 h1;
(4) FFT get length 256*2 frequency response H1.

The problem comes: Since H is band-limited to (1+alpha)/Ts (alpha is
roll-off factor of shaping filter), the coefficients of H1 are zeros
between (1+alpha)/Ts and 2/Ts. But after padding zeros and FFT the
resulting H1 has nonzero  coefficients between (1+alpha)/Ts and 2/Ts. In
fact it fluctuate in the region.

If we do symbol rate sampling the result is good because H are generally
nonzero, and my simulation shows that no error occurs. While aforementioned
channel estimation method brings error even in the absent of noise.

Anyone can point out which step is wrong?

"buaawh" <eewuhao@gmail.com> writes:

> This problem comes from my simulation of Single Carrier Frequency Domain
> Equalization(SC-FDE) system.
>
> Suppose the symbol rate is Ts, for reason of pulse shaping filtering I
> perform fractional space sampling with sample rate Ts/2. A FFT block length
> is 512 and the pilot(used for channel estimation) length is 64. Channel
> estimation is done as follows:
>
> (1) use length 64*2(sampling rate Ts/2) pilot data to estimation channel
> frequency response H;
> (2) IFFT transforms H to time response h;
> (3) padding h with zeros to length 256*2 h1;
> (4) FFT get length 256*2 frequency response H1.
>
> The problem comes: Since H is band-limited to (1+alpha)/Ts (alpha is
> roll-off factor of shaping filter), the coefficients of H1 are zeros
> between (1+alpha)/Ts and 2/Ts. But after padding zeros and FFT the
> resulting H1 has nonzero  coefficients between (1+alpha)/Ts and 2/Ts. In
> fact it fluctuate in the region.
>
> If we do symbol rate sampling the result is good because H are generally
> nonzero, and my simulation shows that no error occurs. While aforementioned
> channel estimation method brings error even in the absent of noise.
>
> Anyone can point out which step is wrong?

Perhaps pondering these questions may lead you to your answer:

1. Do you already have timing and carrier offset at the point you
perform the H FFT?

2. Are you using raised-cosine filtering? If so, then your H spectrum is
not the channel response but the channel response cascaded with the
transmitter's root cosine raised filter. You would have to remove the
RRC response to obtain the actual channel response.

3. Just hacking off the input data at 64*2 samples is going to convolve
your actual channel response with a sinc-like function.

4. Why are you sampling at Ts? You state up front the sample rate
(period) is Ts/2.
-- 
%  Randy Yates                  % "My Shangri-la has gone away, fading like 
%% Fuquay-Varina, NC            %  the Beatles on 'Hey Jude'" 
%%% 919-577-9882                %  
%%%% <yates@ieee.org>           % 'Shangri-La', *A New World Record*, ELO
http://www.digitalsignallabs.com

>"buaawh" <eewuhao@gmail.com> writes:
>
>> This problem comes from my simulation of Single Carrier Frequency
Domain
>> Equalization(SC-FDE) system.
>>
>> Suppose the symbol rate is Ts, for reason of pulse shaping filtering I
>> perform fractional space sampling with sample rate Ts/2. A FFT block
length
>> is 512 and the pilot(used for channel estimation) length is 64.
Channel
>> estimation is done as follows:
>>
>> (1) use length 64*2(sampling rate Ts/2) pilot data to estimation
channel
>> frequency response H;
>> (2) IFFT transforms H to time response h;
>> (3) padding h with zeros to length 256*2 h1;
>> (4) FFT get length 256*2 frequency response H1.
>>
>> The problem comes: Since H is band-limited to (1+alpha)/Ts (alpha is
>> roll-off factor of shaping filter), the coefficients of H1 are zeros
>> between (1+alpha)/Ts and 2/Ts. But after padding zeros and FFT the
>> resulting H1 has nonzero  coefficients between (1+alpha)/Ts and 2/Ts.
In
>> fact it fluctuate in the region.
>>
>> If we do symbol rate sampling the result is good because H are
generally
>> nonzero, and my simulation shows that no error occurs. While
aforementioned
>> channel estimation method brings error even in the absent of noise.
>>
>> Anyone can point out which step is wrong?
>
>Perhaps pondering these questions may lead you to your answer:
>
>1. Do you already have timing and carrier offset at the point you
>perform the H FFT?
>
>2. Are you using raised-cosine filtering? If so, then your H spectrum is
>not the channel response but the channel response cascaded with the
>transmitter's root cosine raised filter. You would have to remove the
>RRC response to obtain the actual channel response.
>
>3. Just hacking off the input data at 64*2 samples is going to convolve
>your actual channel response with a sinc-like function.
>
>4. Why are you sampling at Ts? You state up front the sample rate
>(period) is Ts/2.
>-- 
>%  Randy Yates                  % "My Shangri-la has gone away, fading
like 
>%% Fuquay-Varina, NC            %  the Beatles on 'Hey Jude'" 
>%%% 919-577-9882                %  
>%%%% <yates@ieee.org>           % 'Shangri-La', *A New World Record*,
ELO
>http://www.digitalsignallabs.com
>

Thanks for your hints on the question, following which I found my mistake:
the impulse response is channel response cascaded with RRC pulse shaping
filter, resulting a precursor and after ifft the precursor coefficients
circularly shift to the last several points. So directly pad zero is wrong.
shifting h=[h1, h2] to h'=[h2,h1] and padding zeros on the boundary solve
it. 

Randy Yates wrote:

   ...

> 4. Why are you sampling at Ts? You state up front the sample rate
> (period) is Ts/2.

I just jumped in here, so please excuse any apparent gaffe. Isn't Ts the 
sampling interval by definition? Or isn't it a sampling interval at all?

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