> >Are you sure this is not the Goertzel algorithm?
> Scott
> >has a page that discusses this algorithm.
>
>
> Sure it is not. The Goertzel algorithm calculates
> sine value by a recursive

Yeah, I misunderstood what you were describing. Sorry


> >
> >Also, here's a dtmf generator that uses Goertzel
>
>http://www.ti.com/sc/docs/psheets/abstract/apps/spra066.htm
> >
>
>
> Are you kidding? This is a decoder algorithm.

No, I'm not kidding, just wrong.

> >http://www.g4dvj.demon.co.uk/dds.htm
> >
>
>
> Thanks for the link. Explanation is perfect.

1 out of 3 pieces of information distributed on the
pic list is decent. :)

> I think that my method resembles phase accumulator
> and table method except
> there is no table. The table is generated on the run
> and this is triangle.
> In my case error of triangle generation is lower,
> because there is no
> restriction to use only table values. Finally, this
> reduced error makes the
> sine wave frequency more stable. Certainly, the
> phase accumulator method is
> more straight-forward and simple, at the cost of
> frequency jitter. In case
> of DTMF this is mainly the frequency what matters,
> because DTMF frequencies
> are very close.

True, butt, if you incorporate the proper
interpolation along with your look up table, then the
jitter effects are mitigated. Without any
interpolation, the sinewave output from the look-up
table is a stair-stepped waveform. The higher the
update frequency, the smaller the steps and the closer
to a sine wave approximation. This 'stair-stepped'
wave form however is a form of interpolation;
specifically zero-order interpolation. You may have
heard of a "zero-order hold" in your studies on
signals and systems (and perhaps z-transforms).
According to the 'theory', as long as you satisfy the
Nyquist sampling criterion, you can exactly reproduce
the synthesized wave form. But I couldn't look at you
with a straight face and say it's possible. The caveat
is that a perfect low-pass filter must be constructed.
DTMF competition is much too cheap for us to insist on
perfect analog filters.

So the next step is first order interpolation. For our
application, this is linear interpolation between
consecutive samples in the table. (For sampling
theory, there are some subtle but important
differences on the interpretation of a first-order
hold.) Which is to say, if you need a sample that is
not (exactly) at one of the sample points in your
table, then you approximate it by using linear
interpolation. For example, suppose you had a table
with 10 samples: f(i) is the function defined for i
=0..9 . Now suppose you need f(2.25)? Well, using
linear interpolation you'd approximate this:

f(2.25) - f(2)    f(3)  - f(2)
-------------- = -------------
  2.25  -   2       3   -   2

f(2.25) ~= f(2) + 0.25*(f(3) -f(2))

>From a frequency or spectral point of view, linear
interpolation's effect on higher harmonics are
approximately the square of the harmonics introduced
by zero-order interpolation (sinc vs. sinc^2). And
since the magnitude of the higher harmonics is less
than one, this is a good thing (e.g. 1/9 is much
smaller than 1/3). The direct implication is that the
stringent low-pass filter constraints associated with
the stair stepped waveform can be relaxed.

>From an intuitive view, you can see how first-order
interpolation improves the table values. For example,
if you had 64 samples of sine wave over one cycle,
then the deviation between linear interpolation and
the 'true value' of the sine wave is quite small. In
fact, an error bound may be placed on it.
Specifically, as I alluded to in another thread, you
can use the Cauchy Remainder Theorem for
interpolation. In essence, this theorem states that
the error is bounded by the n'th derivative of the
function your approximating, where n is the order of
the interpolation (1 for linear). So in the above
example,

The error in approximating:

f(2.25) ~= f(2) + 0.25*(f(3) -f(2))

error <= abs(f'(3) - f'(2))

Now there are additional caveats, like you still have
to satisfy Nyquist.

.lo

ps. Please forgive any mistakes, this is all from memory...

=====

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