Relationship of Basis Functions and Sampling Rate

[hrh1818]

According to Steven Smith's book Chapter 9 page 174 "Digital Signal
Processing" a sine wave with a frequency exactly equal to a basis
function will have a sharp peak and no spectral leakage. Whereas a
sine wave with a frequency between two basis functions will have
spectral leakage. If one knows ahead of time the frequencies at which
peaks will occur in a spectrum, say a CFL's current, how should one
choose the sampling rate? Does the data need to be processed with a
window to minimize spectral leakage before performing a DFT if the
sampling rate is an integer multiple of the fundamental and its
harmonics?

Howard

[Tim Wescott]

On Mon, 05 Oct 2009 08:18:08 -0700, hrh1818 wrote:

> According to Steven Smith's book Chapter 9 page 174 "Digital Signal
> Processing" a sine wave with a frequency exactly equal to a basis
> function will have a sharp peak and no spectral leakage. Whereas a sine
> wave with a frequency between two basis functions will have spectral
> leakage. If one knows ahead of time the frequencies at which peaks will
> occur in a spectrum, say a CFL's current, how should one choose the
> sampling rate? Does the data need to be processed with a window to
> minimize spectral leakage before performing a DFT if the sampling rate
> is an integer multiple of the fundamental and its harmonics?
>
> Howard

For a perfectly periodic signal* the DFT is exact, and no windowing is
necessary. You need windowing because the DFT 'expects' a periodic, and
gives confusing results when handed a signal with discontinuities at the
ends.

So, if you have a perfectly periodic signal then all of your basis
functions will be perfect harmonics of the fundamental, and a DFT will be
exact**. So windowing will be the _wrong_ thing to do, instead of
recommended.

* Or one that exists on a timeline that is circular, if and only if that
statement suppresses the inevitable flame war.

** Barring numerical difficulties. Put that flame thrower _away_.

--
www.wescottdesign.com

[Jerry Avins]

hrh1818 wrote:
> According to Steven Smith's book Chapter 9 page 174 "Digital Signal
> Processing" a sine wave with a frequency exactly equal to a basis
> function will have a sharp peak and no spectral leakage. Whereas a
> sine wave with a frequency between two basis functions will have
> spectral leakage. If one knows ahead of time the frequencies at which
> peaks will occur in a spectrum, say a CFL's current, how should one
> choose the sampling rate? Does the data need to be processed with a
> window to minimize spectral leakage before performing a DFT if the
> sampling rate is an integer multiple of the fundamental and its
> harmonics?

Your use of "basis function" shows that you need to go back and do more
reading.

The condition for no leakage is more stringent than you suppose. The
samped waveform must include an integer number of cycles of each of its
components frequencies.

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

[joepierson]

On Oct 5, 11:18*am, hrh1818 <[email protected]> wrote:
> According to Steven Smith's book Chapter 9 page 174 "Digital Signal
> Processing" *a sine wave *with a frequency exactly equal to a basis
> function will have a sharp peak and no spectral leakage. *Whereas a
> sine wave with a frequency between two basis functions will have
> spectral leakage. If one knows ahead of time the frequencies at which
> peaks will occur in a spectrum, say a CFL's current, how should one
> choose the sampling rate? *Does the data need to be processed with a
> window to minimize spectral leakage before performing a DFT if the
> sampling rate is an integer multiple of the fundamental and its
> harmonics?
>
> Howard

To minimize spectral leakage I (and many other people) use a flat top
window

http://www.diracdelta.co.uk/science/source/f/l/flattop%20window/source.html#

It will give you the correct magnitude answer no matter where the
signal frequency is respective to the basis functions frequencies. The
downside is resolution is lost. As long as you keep that in mind when
looking at the results, it is a very useful window if you need precise
magnitude measurements.

[Rune Allnor]

On 5 Okt, 17:18, hrh1818 <[email protected]> wrote:
> According to Steven Smith's book Chapter 9 page 174 "Digital Signal
> Processing" *a sine wave *with a frequency exactly equal to a basis
> function will have a sharp peak and no spectral leakage. *Whereas a
> sine wave with a frequency between two basis functions will have
> spectral leakage. If one knows ahead of time the frequencies at which
> peaks will occur in a spectrum, say a CFL's current, how should one
> choose the sampling rate?

One doesn't care. One selects the sampling rate such that the
Nyquist sampling criterion is satisfied with some margin, and
goes on from there. If one wants a particular resolution of
the spectrum, one selects a duration of the observation such
that the time-bandwidth product plays in your favour.

>*Does the data need to be processed with a
> window to minimize spectral leakage before performing a DFT if the
> sampling rate is an integer multiple of the fundamental and its
> harmonics?

There might be reasons to do that, but there are no such
requirements what the DFT is concerned.

Rune

[Dale Dalrymple]

On Oct 5, 9:27 am, joepierson <[email protected]> wrote:
> On Oct 5, 11:18 am, hrh1818 <[email protected]> wrote:
> ...
> To minimize spectral leakage I (and many other people) use a flat top
> window
>
> http://www.diracdelta.co.uk/science/source/f/l/flattop%20window/sourc...
>
> It will give you the correct magnitude answer no matter where the
> signal frequency is respective to the basis functions frequencies. The
> downside is resolution is lost. As long as you keep that in mind when
> looking at the results, it is a very useful window if you need precise
> magnitude measurements.

Flattop windows can give accurate amplitude response for tones with
high SNR that are well separated from other tones. That is not what is
usually meant by 'reducing leakage' as other bin responses don't go to
zero.

Actually, all windows have 'leakage' but if you have a single sine
wave with an integer number of cycles in a rectangular window and
calculate a DFT with the same size as the window, one bin will sample
the peak of the sine wave's frequency response and the rest of the
bins will sample zeros of the 'leakage function'. If you zero extend
and calculate a DFT of twice the length of the window, half the bins
will have the same response as the original DFT and half will have non-
zero responses sampling the 'leakage function' between it's zeros.

Dale B. Dalrymple

[Fred Marshall]

Rune Allnor wrote:
> On 5 Okt, 17:18, hrh1818 <[email protected]> wrote:
>> According to Steven Smith's book Chapter 9 page 174 "Digital Signal
>> Processing" a sine wave with a frequency exactly equal to a basis
>> function will have a sharp peak and no spectral leakage. Whereas a
>> sine wave with a frequency between two basis functions will have
>> spectral leakage. If one knows ahead of time the frequencies at which
>> peaks will occur in a spectrum, say a CFL's current, how should one
>> choose the sampling rate?
>
> One doesn't care. One selects the sampling rate such that the
> Nyquist sampling criterion is satisfied with some margin, and
> goes on from there. If one wants a particular resolution of
> the spectrum, one selects a duration of the observation such
> that the time-bandwidth product plays in your favour.
>
>> Does the data need to be processed with a
>> window to minimize spectral leakage before performing a DFT if the
>> sampling rate is an integer multiple of the fundamental and its
>> harmonics?
>
> There might be reasons to do that, but there are no such
> requirements what the DFT is concerned.
>
> Rune

There may be a "requirement" related to SNR. That's why we sometimes
clock with a tachometer - so that the samples are tied to the rotational
position of a machine. This yields higher SNR of tonals than if one
uses an arbitrary clock. That's pretty much the same idea.

If the clock is arbitrary then the spectral leakage may add to the
apparent SNR as certainly does wow and flutter, jitter, etc.
If the clock is "locked" to the signals then there is no wow, flutter or
jitter and spectral leakage can usually be eliminated and the SNR can be
better - particularly at higher resolutions I should think.

I don't like saying "basis functions" when one means a sinusoid in
particular. There are others including sincs, etc. So, it's best to
say what you mean.

Fred

[glen herrmannsfeldt]

Tim Wescott <[email protected]> wrote:
< On Mon, 05 Oct 2009 08:18:08 -0700, hrh1818 wrote:

<> According to Steven Smith's book Chapter 9 page 174 "Digital Signal
<> Processing" a sine wave with a frequency exactly equal to a basis
<> function will have a sharp peak and no spectral leakage. Whereas a sine
<> wave with a frequency between two basis functions will have spectral
<> leakage. If one knows ahead of time the frequencies at which peaks will
<> occur in a spectrum, say a CFL's current, how should one choose the
<> sampling rate? Does the data need to be processed with a window to
<> minimize spectral leakage before performing a DFT if the sampling rate
<> is an integer multiple of the fundamental and its harmonics?

Use a PLL and lock onto a multiple of the fundamental.

< For a perfectly periodic signal* the DFT is exact, and no windowing is
< necessary. You need windowing because the DFT 'expects' a periodic, and
< gives confusing results when handed a signal with discontinuities at the
< ends.

If the signal and DFT have the same period. It seems that the OP
has cases where that isn't true. It should be easier to fix that
than to fix the DFT.

-- glen

[glen herrmannsfeldt]

Fred Marshall <fmarshallx@remove_the_xacm.org> wrote:
(snip)

< I don't like saying "basis functions" when one means a sinusoid in
< particular. There are others including sincs, etc. So, it's best to
< say what you mean.

The basis functions of the Fourier transforms are sinusoids.

There is, for example, the Fourier-Bessel transform used in
cylindrical coordinates which has sinusoids for some variables
and bessel functions for the radial basis functions.

-- glen

[Tim Wescott]

On Mon, 05 Oct 2009 18:42:09 +0000, glen herrmannsfeldt wrote:

> Tim Wescott <[email protected]> wrote: < On Mon, 05 Oct 2009 08:18:08
> -0700, hrh1818 wrote:
>
> <> According to Steven Smith's book Chapter 9 page 174 "Digital Signal
> <> Processing" a sine wave with a frequency exactly equal to a basis
> <> function will have a sharp peak and no spectral leakage. Whereas a
> sine <> wave with a frequency between two basis functions will have
> spectral <> leakage. If one knows ahead of time the frequencies at which
> peaks will <> occur in a spectrum, say a CFL's current, how should one
> choose the <> sampling rate? Does the data need to be processed with a
> window to <> minimize spectral leakage before performing a DFT if the
> sampling rate <> is an integer multiple of the fundamental and its
> harmonics?
>
> Use a PLL and lock onto a multiple of the fundamental.
>
> < For a perfectly periodic signal* the DFT is exact, and no windowing is
> < necessary. You need windowing because the DFT 'expects' a periodic,
> and < gives confusing results when handed a signal with discontinuities
> at the < ends.
>
> If the signal and DFT have the same period.

Hmm. I was thinking that _real hard_, but I didn't get it on the page...

> It seems that the OP has
> cases where that isn't true. It should be easier to fix that than to
> fix the DFT.

Agreed.

--
www.wescottdesign.com

[robert bristow-johnson]

On Oct 5, 12:33*pm, Rune Allnor <[email protected]> wrote:
> On 5 Okt, 17:18, hrh1818 <[email protected]> wrote:
> ...
> >*Does the data need to be processed with a
> > window to minimize spectral leakage before performing a DFT if the
> > sampling rate is an integer multiple of the fundamental and its
> > harmonics?
>
> There might be reasons to do that, but there are no such
> requirements what the DFT is concerned.

i know this was controversial with some (and still do not believe it
to be intrinsically controversial), but think of the DFT as a mapping
of one discrete, infinite, and periodic sequence with period N to
another discrete, infinite, and periodic sequence with the same
period. and the iDFT does the same thing but maps it back.

what this means is that the DFT theoretically periodically extends the
finite-lengthed (of length N) sequence that is passed to it. it
always "assumes" that the N samples passed to it comprise exactly one
periodic of this periodic sequence of period N.

so, if the number of samples per period is precisely an integer, N,
and that is the same as the DFT length, they effect of yanking out
those N numbers and passing those N numbers to the DFT means that the
inherent periodic extension that the DFT does changes nothing. if
that is not the case, the effect of windowing is what happens when you
yank out N samples out of this sequence that is not periodic with
period N.

r b-j

[hrh1818]

On Oct 5, 10:51*am, Jerry Avins <[email protected]> wrote:

>
> Your use of "basis function" shows that you need to go back and do more
> reading.
>
> The condition for no leakage is more stringent than you suppose. The
> samped waveform must include an integer number of cycles of each of its
> components frequencies.

Thanks Jerry, You confirmed my suspicion that it is practically
impossible to judiciously select the sampling frequency to eliminate
the need for a window. For example for the 3, 5, 7, 9, 11, 13, 15,
17, and 19 odd harmonics would require a sampling frequency of
43,648,605 samples per second.to collect an integer number of cycles
for all of these odd harmonics.

Howard


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

[glen herrmannsfeldt]

hrh1818 <[email protected]> wrote:
(snip)

< Thanks Jerry, You confirmed my suspicion that it is practically
< impossible to judiciously select the sampling frequency to eliminate
< the need for a window. For example for the 3, 5, 7, 9, 11, 13, 15,
< 17, and 19 odd harmonics would require a sampling frequency of
< 43,648,605 samples per second.to collect an integer number of cycles
< for all of these odd harmonics.

Hmmm. If you sample one cycle of the fundamental that will contain
an integer number of cycles of any harmonic. I don't see why you
want the LCM of them. An integer number of samples for the fundamental,
easy with a PLL, hard without one, should do it. (or, I suppose,
now a DLL or otherwise locking onto the source.)

-- glen

[Jerry Avins]

hrh1818 wrote:
> On Oct 5, 10:51 am, Jerry Avins <[email protected]> wrote:
>
>> Your use of "basis function" shows that you need to go back and do more
>> reading.
>>
>> The condition for no leakage is more stringent than you suppose. The
>> samped waveform must include an integer number of cycles of each of its
>> components frequencies.
>
> Thanks Jerry, You confirmed my suspicion that it is practically
> impossible to judiciously select the sampling frequency to eliminate
> the need for a window. For example for the 3, 5, 7, 9, 11, 13, 15,
> 17, and 19 odd harmonics would require a sampling frequency of
> 43,648,605 samples per second.to collect an integer number of cycles
> for all of these odd harmonics.

Two full cycles of second harmonic fit into one cycle of the
fundamental; three cycles of the third, 17 od the 17th, and so on. Every
signal to be sampled has to be bandlimited. If your signal contains 25
harmonics, you need to sample somewhat faster than 50 times the
fundamental frequency. That's not as bad as you seem to think.

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

[Phil O. Sopher]

"joepierson" <[email protected]> wrote in message
news:05bc0357-f340-4f19-a7af-c7f3b20a1c5e@m33g2000pri.googlegroups.com...

> http://www.diracdelta.co.uk/science/source/f/l/flattop%20window/source.html#

Always a joy to discover another previously-unknown encyclopaedia
("encyclopedia" to the Yanks) !

[hrh1818]

On Oct 5, 10:44*pm, Jerry Avins <[email protected]> wrote:
> hrh1818 wrote:
> > On Oct 5, 10:51 am, Jerry Avins <[email protected]> wrote:
>
> >> Your use of "basis function" shows that you need to go back and do more
> >> reading.
>
> >> The condition for no leakage is more stringent than you suppose. The
> >> samped waveform must include an integer number of cycles of each of its
> >> components frequencies.
>
> > Thanks Jerry, * You confirmed my suspicion that it is practically
> > impossible to judiciously select the sampling frequency to eliminate
> > the need for a window. *For example for the *3, 5, 7, 9, 11, 13, 15,
> > 17, and 19 odd harmonics would require *a sampling frequency of
> > 43,648,605 samples per second.to collect an integer number of cycles
> > for all of these *odd harmonics.
>
> Two full cycles of second harmonic fit into one cycle of the
> fundamental; three cycles of the third, 17 od the 17th, and so on. Every
> signal to be sampled has to be bandlimited. If your signal contains 25
> harmonics, you need to sample somewhat faster than 50 times the
> fundamental frequency. That's not as bad as you seem to think.
>
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Jerry, thanks for keeping me from going off in the wrong direction.
There appears to another complication when processes the data with a
FFT. That is the need to collect 2^N samples for an integer number of
cycles of the fundamental. This means the ratio of sampling frequency
to fundamental frequency would need to be:
(2^N samples/number of cycles of fundamental frequency in which data
is collected)
Hence for the application that initiated the original question,
spectral analysis of power line current, trying to judiciously select
the sampling frequency to avoid the need to window the data adds to
much complication.

Howard

[Dale Dalrymple]

On Oct 6, 12:39 am, "Phil O. Sopher" <[email protected]> wrote:
> "joepierson" <[email protected]> wrote in message
>
> news:05bc0357-f340-4f19-a7af-c7f3b20a1c5e@m33g2000pri.googlegroups.com...
>
> >http://www.diracdelta.co.uk/science/source/f/l/flattop%20window/sourc...
>
> Always a joy to discover another previously-unknown encyclopaedia
> ("encyclopedia" to the Yanks) !

It would be a more joyous discovery if they could have gotten
something as widely known as the Kaiser-Bessel window right.

http://www.diracdelta.co.uk/science/source/k/a/kaiser%20bessel%20window/source.html

Dale B. Dalrymple

[Tim Wescott]

On Tue, 06 Oct 2009 09:26:29 -0700, hrh1818 wrote:

> On Oct 5, 10:44Â*pm, Jerry Avins <[email protected]> wrote:
>> hrh1818 wrote:
>> > On Oct 5, 10:51 am, Jerry Avins <[email protected]> wrote:
>>
>> >> Your use of "basis function" shows that you need to go back and do
>> >> more reading.
>>
>> >> The condition for no leakage is more stringent than you suppose. The
>> >> samped waveform must include an integer number of cycles of each of
>> >> its components frequencies.
>>
>> > Thanks Jerry, Â* You confirmed my suspicion that it is practically
>> > impossible to judiciously select the sampling frequency to eliminate
>> > the need for a window. Â*For example for the Â*3, 5, 7, 9, 11, 13, 15,
>> > 17, and 19 odd harmonics would require Â*a sampling frequency of
>> > 43,648,605 samples per second.to collect an integer number of cycles
>> > for all of these Â*odd harmonics.
>>
>> Two full cycles of second harmonic fit into one cycle of the
>> fundamental; three cycles of the third, 17 od the 17th, and so on.
>> Every signal to be sampled has to be bandlimited. If your signal
>> contains 25 harmonics, you need to sample somewhat faster than 50 times
>> the fundamental frequency. That's not as bad as you seem to think.
>>
>> Jerry
>> --
>> Engineering is the art of making what you want from things you can get.
>> ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
>
> Jerry, thanks for keeping me from going off in the wrong direction.
> There appears to another complication when processes the data with a
> FFT. That is the need to collect 2^N samples for an integer number of
> cycles of the fundamental. This means the ratio of sampling frequency to
> fundamental frequency would need to be: (2^N samples/number of cycles of
> fundamental frequency in which data is collected)
> Hence for the application that initiated the original question, spectral
> analysis of power line current, trying to judiciously select the
> sampling frequency to avoid the need to window the data adds to much
> complication.
>
> Howard

Eh???

You mentioned the 19th harmonic -- OK, if you want to measure the 19th
harmonic energy on a 60Hz power line, you need to sample faster than 2 *
19 * 60. Assuming that you're going to use anti-alias filtering, you
probably want to sample at greater than 4 * 19 * 60. Just to make the
math easy you can call it 5 * 20 * 60Hz, or 6000kHz sampling.

6kHz ain't all that fast. It's a bit much if you want to do it with a
$0.25 processor, but for one or two bucks in quantity you can get a hot-
rod 8-bit processor with enough speed to phase lock to the power line,
collect 6000 samples, then analyze the sample vector at leisure.

--
www.wescottdesign.com

[Jerry Avins]

hrh1818 wrote:
> On Oct 5, 10:44 pm, Jerry Avins <[email protected]> wrote:
>> hrh1818 wrote:
>>> On Oct 5, 10:51 am, Jerry Avins <[email protected]> wrote:
>>>> Your use of "basis function" shows that you need to go back and do more
>>>> reading.
>>>> The condition for no leakage is more stringent than you suppose. The
>>>> samped waveform must include an integer number of cycles of each of its
>>>> components frequencies.
>>> Thanks Jerry, You confirmed my suspicion that it is practically
>>> impossible to judiciously select the sampling frequency to eliminate
>>> the need for a window. For example for the 3, 5, 7, 9, 11, 13, 15,
>>> 17, and 19 odd harmonics would require a sampling frequency of
>>> 43,648,605 samples per second.to collect an integer number of cycles
>>> for all of these odd harmonics.
>> Two full cycles of second harmonic fit into one cycle of the
>> fundamental; three cycles of the third, 17 od the 17th, and so on. Every
>> signal to be sampled has to be bandlimited. If your signal contains 25
>> harmonics, you need to sample somewhat faster than 50 times the
>> fundamental frequency. That's not as bad as you seem to think.
>>
>> Jerry
>> --
>> Engineering is the art of making what you want from things you can get.
>> ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
>
> Jerry, thanks for keeping me from going off in the wrong direction.
> There appears to another complication when processes the data with a
> FFT. That is the need to collect 2^N samples for an integer number of
> cycles of the fundamental. This means the ratio of sampling frequency
> to fundamental frequency would need to be:
> (2^N samples/number of cycles of fundamental frequency in which data
> is collected)
> Hence for the application that initiated the original question,
> spectral analysis of power line current, trying to judiciously select
> the sampling frequency to avoid the need to window the data adds to
> much complication.

Windowing isn't so terrible to do, and results without it at arbitrary
sampling rates are quite serviceable for many purposes. Still, there's
no question that synchronous sampling simplifies the thought process.

Most power-line measurements seek to include at least the first 15
harmonics. Because one must sample /faster/ than twice the bandwidth ans
synchronous sampling requires an integer ratio, you would sample at 60
times the line frequency. That gives a generous oversample ratio of 2.
Personally, I'd work back from the highest sampling raye my system could
easily run at, and work back from there to see how many harmonics I
could include. An audio ADC running at 48Ksps could synchronously sample
200 harmonics of a 60Hz line. To make the anti-alias filter simple, I'd
likely design the system for 50 *potential* harmonics and use however
many made sense.

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

[Fred Marshall]

glen herrmannsfeldt wrote:
> Fred Marshall <fmarshallx@remove_the_xacm.org> wrote:
> (snip)
>
> < I don't like saying "basis functions" when one means a sinusoid in
> < particular. There are others including sincs, etc. So, it's best to
> < say what you mean.
>
> The basis functions of the Fourier transforms are sinusoids.
>
> There is, for example, the Fourier-Bessel transform used in
> cylindrical coordinates which has sinusoids for some variables
> and bessel functions for the radial basis functions.
>
> -- glen

Right. And, if that was the context of Smith's book quote then fine.

Otherwise the DFT wasn't mentioned in the OP and one can construct using
sincs as basis functions just as well as sinusoids - as long as one is
dealing with bandlimited or timelimited functions - which is our typical
assumption.

Fred