stenasc@yahoo.com (Bob) wrote in message news:<20540d3a.0307180405.7796b21a@posting.google.com>...
> Hi all,
> 
> Just wondering about the dc component of an fft.

I think a more correct term to use than "dc component" will be "sample
mean".  When a signal has a DC component, there will be an impulse at
zero frequency; however, when the signal has non-zero sample mean, the
transform value at zero frequency value will be non-zero.  This does
not mean the signal has a DC component.  Your question of why you see
a non-zero value has been explained very nicely by others.

And don't call it "... of an fft".  FFT is just a fast algorithm for
computing the DFT; at the end of it you get DFT coefficients, _not_
FFT coefficients.  But terms like FFT coefficients, FFT spectrum, etc.
are so well entrenched in the literature that it is a battle that has
been lost long time ago.  Sigh.

"Jim Thomas" <jthomas@bittware.com> wrote in message
news:3F1801ED.5F06CBA4@bittware.com...
> Bob wrote:
> >
> > Hi all,
> >
> > Just wondering about the dc component of an fft.
> >
> > If I have an input signal e.g a single cosine wave. When I sample it
> > for say a 128 point fft, should it have a dc component value.
> > In my fft, I sum the input samples and calculate the average.
> >
> > I half expected it to be zero (no dc component), but this wasn't the
case.
> >
> > Can anyone shed some light upon this ?
> >
> > If the dc component of said cosine wave isn't zero, what effect will it
> > have on the fft result ?
> >
> > As always...thanks to everyone who replies
> >
> > Bob
>
> If you have an integer number of periods in your cosine, then the DC
> component will be zero.  If you have say 4.8 periods of a cosine in your
> data block, you will get some DC.  This is because the cosine frequency
> does not fall on a "bin center" (as it is often called).  The DC
> component is really just the average of all the samples.  The positive
> part of the cosine will be exactly cancelled by the negative part, but
> if the cosine ends before a period completes, the cancellation is
> incomplete.
>
> When the input frequency does not fall on a bin center, you'll get
> leakage into frequency bins other than DC too.

Talking about bins is one way to do it.  So, I'm not disagreeing.

It is simpler to think in terms of the dc term of the transform being the
average.  As others have said, if the number of periods in the epoch taken
isn't an integer then there won't be zero average.

This gets a little more complicated if you're doing a Discrete Fourier
Transform.  A discrete Fourier Transform or the Fast version: FFT will be
made up of a number of samples.  The span of those samples will be one
sample interval short of a full period.  Like this:

The temporal epoch you have in mind is T.
The number of samples you have in mind is N.
The sample interval is T/N.
The temporal epoch is spanned by N+1 samples.
The temporal epoch contains N sample intervals.
(The sampling frequency is N/T and the frequency sampling interval is 1/T)
There are N samples in frequency spanning (N-1)/T
(There are N+1 samples in frequency spanning N/T Hz).
N time samples span N-1 sample intervals (then it is assumed to repeat with
x(0)=x(N)).
The period spans T seconds and N sample intervals.

So, if:
- the sample interval is 1 second,
- the sample epoch is 128 seconds
- there will be 128 samples spanning 127 seconds and repeating at 128
seconds intervals.
- if the first sample is at time zero, the last sample is at time 127
seconds.

So, if:
- you define a signal to be cos(2*pi*0.1*t),
- you sample it at intervals of 1 second,
- you take 100 samples
- the last sample is at 99 seconds, not 100 seconds.
The average is zero unless you repeat the first sample as the last.
That's easy to see if you sample a single period of a cosine at intervals of
pi/2.
The samples would be 1, 0, -1, 0, 1 which has an average of 0.2
The average of 1, 0 , -1, 0 has an average of zero - the first and last
samples aren't the same but it represents one period nonetheless.

Here I've viewed spans of time and frequency as starting and ending on a
sample.
If you view each sample as "representing" the sample instant and all of the
time thereafter up to but not inucluding the next sample, then the number of
samples and the number of intervals coincides.  Maybe that's a better
way.....

Fred

Bob wrote:
> 
> Hi all,
> 
> Just wondering about the dc component of an fft.
> 
> If I have an input signal e.g a single cosine wave. When I sample it
> for say a 128 point fft, should it have a dc component value.
> In my fft, I sum the input samples and calculate the average.
> 
> I half expected it to be zero (no dc component), but this wasn't the case.
> 
> Can anyone shed some light upon this ?
> 
> If the dc component of said cosine wave isn't zero, what effect will it
> have on the fft result ?
> 
> As always...thanks to everyone who replies
> 
> Bob

By definition, the DC component of a FULL CYCLE of a sinusoid is zero.
This is a cosine sampled six times per cycle at an arbitrary phase:
... 1, 1, 0, -1, -1, 0, .... For a full cycle, the average is clearly
zero. The DC value of half a cycle depends on how the half is chosen. Of
the six possible ways, two are zero, two are 2/3, and two are -2/3. I
can add more, but I trust you can take it from here.

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

Bob wrote:
> 
> Hi all,
> 
> Just wondering about the dc component of an fft.
> 
> If I have an input signal e.g a single cosine wave. When I sample it
> for say a 128 point fft, should it have a dc component value.
> In my fft, I sum the input samples and calculate the average.
> 
> I half expected it to be zero (no dc component), but this wasn't the case.
> 
> Can anyone shed some light upon this ?
> 
> If the dc component of said cosine wave isn't zero, what effect will it
> have on the fft result ?
> 
> As always...thanks to everyone who replies
> 
> Bob

If you have an integer number of periods in your cosine, then the DC
component will be zero.  If you have say 4.8 periods of a cosine in your
data block, you will get some DC.  This is because the cosine frequency
does not fall on a "bin center" (as it is often called).  The DC
component is really just the average of all the samples.  The positive
part of the cosine will be exactly cancelled by the negative part, but
if the cosine ends before a period completes, the cancellation is
incomplete.

When the input frequency does not fall on a bin center, you'll get
leakage into frequency bins other than DC too.

-- 
Jim Thomas            Principal Applications Engineer  Bittware, Inc
jthomas@bittware.com  http://www.bittware.com          (703) 779-7770
In theory, theory and practice are the same, but in practice, they're
not

Bob wrote:

> Hi all,
> 
> Just wondering about the dc component of an fft.
> 
> If I have an input signal e.g a single cosine wave. When I sample it
> for say a 128 point fft, should it have a dc component value. 
> In my fft, I sum the input samples and calculate the average.
> 
> I half expected it to be zero (no dc component), but this wasn't the case.
> 
> Can anyone shed some light upon this ?
> 
> If the dc component of said cosine wave isn't zero, what effect will it
> have on the fft result ?
> 
> As always...thanks to everyone who replies
> 
> Bob

Bob:

Mathematically speaking, there should be no DC component for cos(x) or 
sin(x) (assuming exactly one period or lots of periods are used). This 
is true for any zero-mean signal.

However, for a discrete FFT, depending on the sampling rate, you often 
get "spectral leakage" where strong spectral components bleed over into 
adjacent frequency bins. This could give you a small, non-zero DC component.

Good luck,
OUP

On 18 Jul 2003 05:05:05 -0700, stenasc@yahoo.com (Bob) wrote:

>Hi all,
>
>Just wondering about the dc component of an fft.
>
>If I have an input signal e.g a single cosine wave. When I sample it
>for say a 128 point fft, should it have a dc component value. 
>In my fft, I sum the input samples and calculate the average.
>
>I half expected it to be zero (no dc component), but this wasn't the case.
>
>Can anyone shed some light upon this ?
>
>If the dc component of said cosine wave isn't zero, what effect will it
>have on the fft result ?
>
>As always...thanks to everyone who replies
>
>Bob

Hi,
  if the 128-sample cosine wave is not an exact integer number 
of cycles, then the average of those 128 samples will not be 
zero, and the DC component of the 128-point FFT will 
not be zero.

Good Luck,
[-Rick-]
 

Hi all,

Just wondering about the dc component of an fft.

If I have an input signal e.g a single cosine wave. When I sample it
for say a 128 point fft, should it have a dc component value. 
In my fft, I sum the input samples and calculate the average.

I half expected it to be zero (no dc component), but this wasn't the case.

Can anyone shed some light upon this ?

If the dc component of said cosine wave isn't zero, what effect will it
have on the fft result ?

As always...thanks to everyone who replies

Bob