The Direct adaptive Frequency Estimation Technique [4] is based on the Least Mean Squares (LMS) algorithm [2]. We introduce an adaptive step size to normalize the result and called the algorithm NDFET [5].
Using trigonometric identities, we can rewrite a sinusoidal signal
of frequency 
as
Letting 
and 
yields
where 
is the estimation error or additive noise.
When 
consists of multiple sinusoids, includes the sum of the
additional sinusoidal terms.
We apply LMS with step size 
to minimize and estimate 
:
From equations (3) and (4), the LMS filter output is
For a fixed step size of 
, the NDFET algorithm in
equation (5) converged within 20 samples (2.5 ms) for each of
the eight single tones used in DTMF signals.
This algorithm can be expanded for signals including more than one sinusoid [5][4]:
We use a periodic adaptive step size
where 
is the period
and 
.
