Hello all,

There is one factor that seems to be missing from texts which describe 
the DFT/FFT (or perhaps I have missed it).  That is: The correlation 
between time domain signal amplitude and frequency domain bin magnitudes.

On the one hand, many DSP libraries are very meticulous about 
documenting the differences between their FFT or IFFT implementation 
results versus Matlab.  For example, one implementation of an N-point 
FFT might require that the results be divided by N to correlate with the 
Matlab results, while another implementation would produce the exact 
results without further calculations.  Even though I don't use Matlab, 
these sorts of issues make sense to me as an expected side effect of 
certain code optimizations.

What I haven't found in the general texts is mention of the values I 
should expect if I plug sin() at a constant frequency into a time domain 
array and calculate the FFT.  Should I expect one frequency magnitude 
bin to have a value of 1.0 because the sin() function varies between 
-1.0 and +1.0? ... or would the magnitude bin be expected to hold a 
value of 2.0 because the peak-to-peak amplitude of sin() is actually 2.0 
(+1.0 - -1.0)?  I am asking specifically about sin() frequencies which 
fall precisely into a frequency bin based on the FFT size.  The general 
case would obviously be more complicated due to potential frequency 
smearing.

The reason I ask is that Apple's Accelerate/vecLib Framework seems to be 
returning 2.0 when I carefully construct time domain test signals with 
'unity' amplitude, and for some reason I was expecting 1.0 ... so what's 
the 'right' answer?

By the way, my test code is taking the additional step of converting the 
complex FFT results from their real and imaginary components into 
magnitude (and phase - which should be optional).



I also have a slightly related question: I assume that if the real and 
imaginary components never exceed +/-1.0, then the magnitude could not 
possibly be greater than 1.4142, but I also have a feeling that the 
nature of the FFT is that the results should probably never produce a 
magnitude greater than 1.0, because the real and imaginary components 
are not completely random - i.e. a given pair of real and imaginary 
components would never both be +1.0 or -1.0 unless something terribly 
unnatural had occurred.  Again, assuming that the time-domain signal 
does not exceed +/-1.0 (and I realize that is not always the case with 
uncontrolled inputs).


Brian Willoughby
Sound Consulting

P.S.  I'm sure that I'll happen across one of my texts that does spell 
this out, now that I've gone to the trouble of writing the question. 
But I do not that I've gone looking for a reference on more than one 
occasion and came up with nothing.

When I was introduced to the continuous Fourier transform and its inverse, =
there was a 1/sqrt(N) in front of each integral. Later, I ran across others=
 in which one integral was left "raw", and the other divided by N. Dividing=
 either the direct or inverse transform works out to be simplest for a part=
icular application, but it seems to me that dividing each by the square roo=
t has only symmetry and aesthetics to recommend it.

Personally, I usually prefer the form where the frequency-domain coefficien=
ts coincide with the peak amplitudes of the components. That is, for a squa=
re wave of amplitude +/-1 and frequency w=3D2*pi*f, the components are
   (4/pi)*[sin(w) + sin(3*w)/3 + sin(5w)/5 + ...].

Needless to say, this applies also to DFT summations.

Jerry
--=20
Engineering is the art of making what you want from things you can get.

On Mar 25, 9:50=A0am, Brian Willoughby
<Sound_Consulting-...@Sounds.wa.com> wrote:
> Hello all,
>
> There is one factor that seems to be missing from texts which describe
> the DFT/FFT (or perhaps I have missed it). =A0That is: The correlation
> between time domain signal amplitude and frequency domain bin magnitudes.
>
> On the one hand, many DSP libraries are very meticulous about
> documenting the differences between their FFT or IFFT implementation
> results versus Matlab. =A0For example, one implementation of an N-point
> FFT might require that the results be divided by N to correlate with the
> Matlab results, while another implementation would produce the exact
> results without further calculations. =A0Even though I don't use Matlab,
> these sorts of issues make sense to me as an expected side effect of
> certain code optimizations.
>
> What I haven't found in the general texts is mention of the values I
> should expect if I plug sin() at a constant frequency into a time domain
> array and calculate the FFT. =A0Should I expect one frequency magnitude
> bin to have a value of 1.0 because the sin() function varies between
> -1.0 and +1.0? ... or would the magnitude bin be expected to hold a
> value of 2.0 because the peak-to-peak amplitude of sin() is actually 2.0
> (+1.0 - -1.0)? =A0I am asking specifically about sin() frequencies which
> fall precisely into a frequency bin based on the FFT size. =A0The general
> case would obviously be more complicated due to potential frequency
> smearing.
>
> The reason I ask is that Apple's Accelerate/vecLib Framework seems to be
> returning 2.0 when I carefully construct time domain test signals with
> 'unity' amplitude, and for some reason I was expecting 1.0 ... so what's
> the 'right' answer?
>
> By the way, my test code is taking the additional step of converting the
> complex FFT results from their real and imaginary components into
> magnitude (and phase - which should be optional).
>
> I also have a slightly related question: I assume that if the real and
> imaginary components never exceed +/-1.0, then the magnitude could not
> possibly be greater than 1.4142, but I also have a feeling that the
> nature of the FFT is that the results should probably never produce a
> magnitude greater than 1.0, because the real and imaginary components
> are not completely random - i.e. a given pair of real and imaginary
> components would never both be +1.0 or -1.0 unless something terribly
> unnatural had occurred. =A0Again, assuming that the time-domain signal
> does not exceed +/-1.0 (and I realize that is not always the case with
> uncontrolled inputs).
>
> Brian Willoughby
> Sound Consulting
>
> P.S. =A0I'm sure that I'll happen across one of my texts that does spell
> this out, now that I've gone to the trouble of writing the question.
> But I do not that I've gone looking for a reference on more than one
> occasion and came up with nothing.

Hello Brian,

You've ventured into an area where things are done in more than one
way. Most commonly done is the DFT(FFT) has a gain of N/2 where is is
the "size" of the DFT(FFT). This gain results when no scale factor is
applied to the direct DFT(FFT) process. Then the inverse DFT(FFT) will
end up being multiplied by 2/N to make the round trip gain equal to
one. But I know of at least one case where both the forward and
inverse DFTs(FFTs) are multiplied by sqrt(2/N) to impart a sort of
symmetry to the computational process. A simple test of transforming a
sinusoid whose frequency is one of the bin frequencies (avoids
leakage) and seeing what the gain is will tell you what gain your
process has.


But from the defn of a DFT and applying it to a sinusoid on bin
frequency will show you where the gain of N/2 comes from.

IHTH,
Clay

On 2011/03/25 07:26, Clay wrote:
> You've ventured into an area where things are done in more than one
> way. Most commonly done is the DFT(FFT) has a gain of N/2 where is is
> the "size" of the DFT(FFT). This gain results when no scale factor is
> applied to the direct DFT(FFT) process. Then the inverse DFT(FFT) will
> end up being multiplied by 2/N to make the round trip gain equal to
> one. But I know of at least one case where both the forward and
> inverse DFTs(FFTs) are multiplied by sqrt(2/N) to impart a sort of
> symmetry to the computational process. A simple test of transforming a
> sinusoid whose frequency is one of the bin frequencies (avoids
> leakage) and seeing what the gain is will tell you what gain your
> process has.
>
>
> But from the defn of a DFT and applying it to a sinusoid on bin
> frequency will show you where the gain of N/2 comes from.

Is that what Matlab produces?

I must admit that I haven't been able to justify the cost of a Matlab 
license, and thus I've never used it.  Yet Apple and Texas Instruments 
seem to define their FFT libraries in terms of how Matlab does things. 
I always assumed that Matlab was producing the results as defined by the 
DFT, regardless of whether it was 'efficient' for the given processor, 
where Apple and TI are interested in implementations which take the 
fewest cycles, and thus may end up with a different scaling.

Yours comments leave me thinking that a sinusoid with an amplitude of 1 
(or a range of +/-1) should have a different bin magnitude for each FFT 
size.


P.S.  Another related question is what to expect from impulse response 
and step response inputs.  I'm inclined to synthesize an impulse as a 
single +1.0 sample followed by 0.0 samples to pad out the FFT size. 
Similarly, a step input would be a 0.0 sample followed by +1.0 samples 
to pad out the FFT size.  In both cases, I would hope that the output is 
limited to +1.0 in each bin magnitude, unless the system under test has 
overshoot, ringing, resonance, or something like that.  e.g., I should 
be able to test the frequency response of an unknown process by sending 
an impulse to the input and then calculating an FFT on the output.  A 
'flat' frequency response would then be expected as  bins with +1.0 at 
every frequency.  Have I got the calibration right?

Brian Willoughby
Sound Consulting

On Mar 27, 9:29=A0am, Brian Willoughby
<Sound_Consulting-...@Sounds.wa.com> wrote:

....
>  =A0Another related question is what to expect from impulse response
> and step response inputs. =A0I'm inclined to synthesize an impulse as a
> single +1.0 sample followed by 0.0 samples to pad out the FFT size.
> Similarly, a step input would be a 0.0 sample followed by +1.0 samples
> to pad out the FFT size. =A0In both cases, I would hope that the output i=
s
> limited to +1.0 in each bin magnitude, unless the system under test has
> overshoot, ringing, resonance, or something like that. =A0e.g., I should
> be able to test the frequency response of an unknown process by sending
> an impulse to the input and then calculating an FFT on the output. =A0A
> 'flat' frequency response would then be expected as =A0bins with +1.0 at
> every frequency. =A0Have I got the calibration right?

Brian,

this is a very similar issue to the one posted by Andy365 in the
thread: "Energy in a signal as a function of sampling".

there is a well-defined answer for both of you.  or maybe a couple of
different answers that will get you to the same place.

the way i dealt with this when i was in college and using the computer
to do FFTs and such on data that (in simulation) came from a known
source, was to compare the Discrete Fourier Transform (however it's
defined, whether it has a 1, a 1/N, or a sqrt(1/N) in front) to a
Riemann Summation of the Fourier Integral.

the other way i can think of is to simply define a unit sinusoid, bust
it into a positive frequency e^(jwn) and a negative frequency
component, e(-jwn), and stuff that into the DFT and see what comes
out.  if w =3D 2*pi*k/N for integer k, it will come out real clean and
you will know how large the number in the FFT bin is for a unit-
amplitude sine.

r b-j

On 2011/03/27 20:21, robert bristow-johnson wrote:
> the way i dealt with this when i was in college and using the computer
> to do FFTs and such on data that (in simulation) came from a known
> source, was to compare the Discrete Fourier Transform (however it's
> defined, whether it has a 1, a 1/N, or a sqrt(1/N) in front) to a
> Riemann Summation of the Fourier Integral.
>
> the other way i can think of is to simply define a unit sinusoid, bust
> it into a positive frequency e^(jwn) and a negative frequency
> component, e(-jwn), and stuff that into the DFT and see what comes
> out.  if w = 2*pi*k/N for integer k, it will come out real clean and
> you will know how large the number in the FFT bin is for a unit-
> amplitude sine.

Thanks for the response.

I hope I don't seem lazy, but I was hoping to just get the answer.  ;-)
I don't mind also knowing how to independently derive said answer.

In college, I remember (thinking) that I understood all of those 
formulae with 'e' and found them to be a beautiful expression of the 
symmetry of nature.  These days, I have forgotten quite a bit of that, 
assuming I really ever had it right.  Now I tend to only understand the 
direct and practical application rather than the theoretical proofs. 
Actually, I've never been able to just look at a formula like e^(j2πf/T) 
and 'know' what its integral would be.  I tend to depend upon the math 
types to tell me what it works out to be.

That said, if I make an attempt to follow what you wrote above, I find 
myself asking why you translate a sinusoid into both e^(jwn) and 
e^(-jwn) ... wouldn't that put double the signal in?  Do you really need 
both the negative and positive frequencies for a real-valued sinusoid 
input?  Why wouldn't a pure real sinusoid simply translate to the 
positive frequency only?  I cracked open Steiglitz' DPS Primer and the 
first example I came across translated sin(kwt) into e^(jkwt), without 
the negative frequency term.  To be precise, he translates sin(kπt/T) 
with period of 2T into e^(jk2πt/T) with period of T, which is a little 
confusing in itself because of the change in period, but there you have it.

In any event, lately I've been finding myself wishing that someone could 
write a book on DSP which uses plain English at a somewhat high level to 
introduce each concept, even if there is still a followup mathematical 
proof to back up the claims.  But I'd like it organized so that I could 
easily skip the proof until I need to know it and understand it.  Every 
text I've seen seems to start with the math, but that ends up stretching 
out the description long enough that it's harder to follow unless you 
already understand the math.  Sort of counterproductive.  I'm not saying 
that some understanding of the math is not useful or even necessary, but 
I also have a feeling that some of these concepts could easily be 
described to 90% of the necessary level of detail without too much math, 
at least by someone gifted in writing (yes, an admittedly rare trait 
among engineers and scientists, not to mention mathematicians).  I think 
Hal Chamberlin's "Musical Applications of Microprocessors" was the last 
time I sat down for some light reading in plain English and found myself 
in an "a ha, this all makes sense on a deeper level, now" kind of 
moment.  Of course, only an engineer would consider that book to be 
light reading, but I find it a lot lighter than the mathematical texts.

Anyway, sorry for the rambling, but my original question remains: Can 
anyone just spell out the answer in simple terms?  Two very helpful 
individuals have replied in this thread, but basically suggested that I 
derive my answer by manually calculating the DFT.  That's an intriguing 
challenge, but my initial goal was to find a confirmed reference that I 
could use to test my code against (and the library code that I am 
using), and thus I feel like if I derive the answer myself then I would 
still have some doubt.  Actually, I'm refining some old code where I 
thought I had all of these calibrations figured out, and now I'm having 
doubts, thus the request for a solid answer.

P.S.  Who is Riemann?  I never heard of him before.  Actually, scratch 
my literal question, because Wikipedia has the answer.  They even 
mention Fourier in the article...

Brian Willoughby
Sound Consulting

On Mar 25, 3:50=A0pm, Brian Willoughby
<Sound_Consulting-...@Sounds.wa.com> wrote:

> What I haven't found in the general texts is mention of the values I
> should expect if I plug sin() at a constant frequency into a time domain
> array and calculate the FFT. =A0Should I expect one frequency magnitude
> bin to have a value of 1.0 because the sin() function varies between
> -1.0 and +1.0? ... or would the magnitude bin be expected to hold a
> value of 2.0 because the peak-to-peak amplitude of sin() is actually 2.0
> (+1.0 - -1.0)? =A0I am asking specifically about sin() frequencies which
> fall precisely into a frequency bin based on the FFT size. =A0The general
> case would obviously be more complicated due to potential frequency
> smearing.

Those are two questions, and thus they have two answers.

First, assume the sinusoidal has a frequency that coincides
with a DFT coefficient. In that case, it is common to define
the DFT such that the DFT coefficient is N times as large as
the amplitude of the sinusoidal, where N is the DFT length.
There is an 1/N assymetry in the scaling coefficinets of the
DFT and IDFT, such that (formally) IDFT(DFT(x[n])) =3D x[n].

The second case is when the frequency of the sinusoidal does
*not* coincide with a DFT coefficnet. In this case all the
issues regarding scaling as considered in case 1 still applies,
along with the frequency smearing issues.

Rune

On 2011/03/28 00:18, Rune Allnor wrote:
> On Mar 25, 3:50 pm, Brian Willoughby
> <Sound_Consulting-...@Sounds.wa.com>  wrote:
>
>> What I haven't found in the general texts is mention of the values I
>> should expect if I plug sin() at a constant frequency into a time domain
>> array and calculate the FFT.  Should I expect one frequency magnitude
>> bin to have a value of 1.0 because the sin() function varies between
>> -1.0 and +1.0? ... or would the magnitude bin be expected to hold a
>> value of 2.0 because the peak-to-peak amplitude of sin() is actually 2.0
>> (+1.0 - -1.0)?  I am asking specifically about sin() frequencies which
>> fall precisely into a frequency bin based on the FFT size.  The general
>> case would obviously be more complicated due to potential frequency
>> smearing.
>
> Those are two questions, and thus they have two answers.
>
> First, assume the sinusoidal has a frequency that coincides
> with a DFT coefficient. In that case, it is common to define
> the DFT such that the DFT coefficient is N times as large as
> the amplitude of the sinusoidal, where N is the DFT length.
> There is an 1/N assymetry in the scaling coefficinets of the
> DFT and IDFT, such that (formally) IDFT(DFT(x[n])) = x[n].
>
> The second case is when the frequency of the sinusoidal does
> *not* coincide with a DFT coefficnet. In this case all the
> issues regarding scaling as considered in case 1 still applies,
> along with the frequency smearing issues.

Thanks.  I realize that it's basically two questions, and I was 
attempting to exclude the second question.

Getting back to it, I think that part of what I am missing is experience 
with Matlab.  Both Apple and Texas Instruments seem intent to describe 
their FFT library in terms of how the results compare to Matlab's 
results.  That's where I get a little lost.

In your answer, you refer to DFT coefficients.  Would that be the same 
as the FFT bins in the result?

You mention that the DFT coefficient is N times as large as the 
amplitude of the sinusoid, but I would interpret that to mean that if I 
send in a +/-1.0 sin() at a nice frequency for a 128-point FFT, then I 
would see +128 in the result bin for that frequency (or maybe +256). 
What's confusing, though, is that Texas Instruments claims their 
unscaled FFT produces 1/N results compared to Matlab.  i.e. you must 
multiply the Matlab results by N to compare to the TI DSPLIB results. 
This seems entirely opposite of what you describe.  Meanwhile, TI says 
that their scaled FFT produces the exact same results as Matlab, which 
doesn't really tell me anything since I'm unfamiliar with Matlab.  The 
reason I'm confused is that your description implies (to me) that I 
would need to divide by N to get the calibrated results that I want, and 
yet TI only mentions multiplying by N or leaving the values alone.  Some 
piece here doesn't jibe with the rest.

P.S.  I assume that, when the input frequency does not coincide exactly 
with a DFT coefficient, the magnitude of the results for unaligned 
signal frequencies will be less than or almost equal to the aligned 
frequency results.  In other words, I assume that the energy of the 
signal is diminished by being spread out over multiple bins.

Brian Willoughby
Sound Consulting

On Mar 28, 9:42=A0am, Brian Willoughby
<Sound_Consulting-...@Sounds.wa.com> wrote:

> You mention that the DFT coefficient is N times as large as the
> amplitude of the sinusoid,

Use x[n] =3D exp(j 2*pi*q/N), q and N integers, q < N/2.

Then (view with fixed-width font)

       N-1
X[k] =3D sum exp(j2pi q/N) exp(-j2pi k/N)
       n=3D0

If q =3D/=3D k then X[k] =3D=3D 0. When q =3D=3D k then

       N-1                                 N-1
X[k] =3D sum exp(j2pi k/N) exp(-j2pi k/N)  =3D sum 1 =3D N.
       n=3D0                                 n=3D0

So the DFT coefficient X[q] has N times as large amplitude
as the sinusoidal x[n].

Do note the 1/N factor in the expression for the IDFT.

Rune

On Mar 28, 3:08=C2=A0am, Brian Willoughby
<Sound_Consulting-...@Sounds.wa.com> wrote:
> On 2011/03/27 20:21, robert bristow-johnson wrote:
>
> > the way i dealt with this when i was in college and using the computer
> > to do FFTs and such on data that (in simulation) came from a known
> > source, was to compare the Discrete Fourier Transform (however it's
> > defined, whether it has a 1, a 1/N, or a sqrt(1/N) in front) to a
> > Riemann Summation of the Fourier Integral.
>
> > the other way i can think of is to simply define a unit sinusoid, bust
> > it into a positive frequency e^(jwn) and a negative frequency
> > component, e(-jwn), and stuff that into the DFT and see what comes
> > out. =C2=A0if w =3D 2*pi*k/N for integer k, it will come out real clean=
 and
> > you will know how large the number in the FFT bin is for a unit-
> > amplitude sine.
>
> Thanks for the response.
>
> I hope I don't seem lazy, but I was hoping to just get the answer. =C2=A0=
;-)
> I don't mind also knowing how to independently derive said answer.
>
> In college, I remember (thinking) that I understood all of those
> formulae with 'e' and found them to be a beautiful expression of the
> symmetry of nature. =C2=A0These days, I have forgotten quite a bit of tha=
t,
> assuming I really ever had it right. =C2=A0Now I tend to only understand =
the
> direct and practical application rather than the theoretical proofs.
> Actually, I've never been able to just look at a formula like e^(j2=CF=80=
f/T)
> and 'know' what its integral would be. =C2=A0

Don't worry, nobody can, not even Euler or Gauss. You can derive it,
but not intitutively do it. (Or one
can pretend by talking in axiomatic circles I suppose)

Anyway, given a sine wave with peak value of 1 Volt, a 4 point DFT of
the sine wave is

(sin(0)*sin(0) + sin(pi/2)*sin(pi/2) + sin(pi)sin(pi) +
sin(3/2pi)*sin(3/2pi))
=3D (0 + 1+ 0 + 1) =3D 2

(sin(0)*cos(0) + sin(pi/2)*cos(pi/2) + sin(pi)cos(pi) +
sin(3/2pi)*cos(3/2pi))
=3D (0 + 0 + 0 + ) =3D  0

bin magnatide =3D sqrt(2^2+0^2)=3D2

so you get 2 for the bin peak voltage result which is a N/2  =3D 4/2 =3D 2
gain

On Mar 25, 9:50=A0am, Brian Willoughby
<Sound_Consulting-...@Sounds.wa.com> wrote:
> Hello all,
>
> There is one factor that seems to be missing from texts which describe
> the DFT/FFT (or perhaps I have missed it). =A0That is: The correlation
> between time domain signal amplitude and frequency domain bin magnitudes.
>
> On the one hand, many DSP libraries are very meticulous about
> documenting the differences between their FFT or IFFT implementation
> results versus Matlab. =A0For example, one implementation of an N-point
> FFT might require that the results be divided by N to correlate with the
> Matlab results, while another implementation would produce the exact
> results without further calculations. =A0Even though I don't use Matlab,
> these sorts of issues make sense to me as an expected side effect of
> certain code optimizations.
>
> What I haven't found in the general texts is mention of the values I
> should expect if I plug sin() at a constant frequency into a time domain
> array and calculate the FFT. =A0Should I expect one frequency magnitude
> bin to have a value of 1.0 because the sin() function varies between
> -1.0 and +1.0? ... or would the magnitude bin be expected to hold a
> value of 2.0 because the peak-to-peak amplitude of sin() is actually 2.0
> (+1.0 - -1.0)? =A0I am asking specifically about sin() frequencies which
> fall precisely into a frequency bin based on the FFT size. =A0The general
> case would obviously be more complicated due to potential frequency
> smearing.
>
> The reason I ask is that Apple's Accelerate/vecLib Framework seems to be
> returning 2.0 when I carefully construct time domain test signals with
> 'unity' amplitude, and for some reason I was expecting 1.0 ... so what's
> the 'right' answer?
>
> By the way, my test code is taking the additional step of converting the
> complex FFT results from their real and imaginary components into
> magnitude (and phase - which should be optional).
>
> I also have a slightly related question: I assume that if the real and
> imaginary components never exceed +/-1.0, then the magnitude could not
> possibly be greater than 1.4142, but I also have a feeling that the
> nature of the FFT is that the results should probably never produce a
> magnitude greater than 1.0, because the real and imaginary components
> are not completely random - i.e. a given pair of real and imaginary
> components would never both be +1.0 or -1.0 unless something terribly
> unnatural had occurred. =A0Again, assuming that the time-domain signal
> does not exceed +/-1.0 (and I realize that is not always the case with
> uncontrolled inputs).
>
> Brian Willoughby
> Sound Consulting
>
> P.S. =A0I'm sure that I'll happen across one of my texts that does spell
> this out, now that I've gone to the trouble of writing the question.
> But I do not that I've gone looking for a reference on more than one
> occasion and came up with nothing.

There are 3 widely used DFT normalizations (1/N,1,1/sqrt(N)).

I prefer the 1st because cos(2*pi*f0*t) will yield 2 spikes
at +/-f0 of height 0.5.

This is the test I use when encountering a new DFT/FFT
program.

Hope this helps.

Greg

Hope this helps.

On 2011/03/28 16:34, Greg Heath wrote:
> There are 3 widely used DFT normalizations (1/N,1,1/sqrt(N)).
>
> I prefer the 1st because cos(2*pi*f0*t) will yield 2 spikes
> at +/-f0 of height 0.5.
>
> This is the test I use when encountering a new DFT/FFT
> program.
>
> Hope this helps.

Thanks, Greg.

It suddenly occurs to me that the Real-input FFT that is common in DSP 
libraries, and which drops the negative frequencies because they're 
symmetric for real-valued inputs, may be combining those two 0.5 spikes 
into a single 1.0 spike...

I'm pretty sure I was already doing this, but I should test complex FFT 
and real FFT separately for calibration purposes.

Brian Willoughby
Sound Consulting