The following figure collects and organizes the basic operators and transforms related to the process of frequency warping. Some of the basic concepts are discused in the following.

This is a preliminary summary and is still open to changes partially due to the comments we are presently collecting. Therefore, if you have any comments or questions please contact the author.

The __frequency warping concept__ is used in many ways in the
literature depending on topics, contexts and authors in question.

In our studies the basic concepts of frequency warping are defined in the following way:

**Frequency warping **of a function H(f) is performed by
applying the unitary warping operator **U** to the function. (The
operator is defined below, refer to the figure).

Some authors have used the concept __axis warping__ in
connection to the **U** operator, however, the axis (f) is
unchanged because the operator is a H-> ~H mapping. Instead of
warping the axis the operator has created a new, unitarily warped
representation ~H for the function H on the same, unchanged axis.
This operation could also be called __frequency mapping__.

Note that the warping of a function will change its local
__density__. __Frequency warping always affects the spectral
density__.

An example of a frequency warped class of orthonormal functions is
the **FAM
class**. The FAM class can be seen as frequency warped complex
exponentials (complex sinusoids) and the corresponding transform as
the frequency warped inverse Fourier transform (S -> ~s).

**Frequency warping transformation** is a process where one
spectral representation on a certain frequency scale (e.g., Hz,
f-domain) and with a certain frequency resolution (most often
uniform) is transformed to another representation on a new frequency
scale (e.g., Bark or ERB-rate scale, v-domain). The new
representation has a uniform frequency resolution on the new scale -
however, it has a nonuniform resolution when observed from the old
scale. The new frequency scale (v-domain) and the old one (f-domain)
are related by a smooth and monotonic warping (mapping) function v(f)
with a positive derivative (dv/df > 0).

Sometimes the term "frequency warping transformation" is shortened
to frequency warping or to frequency transformation. The essential
difference between the __frequency warping__ and the __frequency
warping transformation__ is that the first one maps the information
back to the same domain f -> f whereas the second transfroms it to
a new domain f -> v.

Typically only band limited signals are considered. Thus both domains f and v are finite. It is practical to normalize the warping function so that the average value of the derivative dv/df is one and if no warping occurs ( v(f)=f ) the derivative equals to one everywhere. In other words, if in some frequency range the derivative is larger than one there must be another range where it is less than one.

The warping function v(f) defines how individual frequency components and different frequency ranges are mapped on the new scale. It also defines how the resoluion of the new representation is allocated, which ranges in the original representation are compressed (shrinked, resolution reduced) and which expanded (stretched, resolution increased). High values of the derivative dv/df indicate a range where the spectral representation is expanded and the resolution is increased. Correspondingly, low values of the derivative indicate the ranges of resolution reduction. At the turning frequency where dv/df=1 the spectral resolution is unchanged.

One important practical application where a warped spectral representation is required is in the computation of auditory spectra ~S(v) from audio signals s(t) or from their Fourier spectra S(f).

As indicated in the following figure there are three main ways a frequency warped spectral representation in the v-domain can be created:

- Frequency warping of the Fourier spectra (FAM analysis & Fourier synthesis, S -> ~S)
- Nonuniform resolution filterbank (s -> ~S )
- Fourier transforming the frequency warped time signal (FAMlet transform & Fourier transform, s-> ~s -> ~S )

The first method when combined to one operation , is equivalent to spectral domain processing with a nonuniform resolution filterbank made of warped (frequency domain) sinc functions.

Which method is chosen depends on the actual practical limitations, e.g., in the implementation of the warping process.

Some details are further discussed in the following notes.

- The unitary warping operator
**U**is primarily defined in the frequency domain. The frequency warping of the time domain signal s(t) can be achieved by a chain of allpass filters. The resulting a-domain corresponds to the tap outputs of the allpass chain. The transformed signal is read simultaneously (at the same time instant) over the taps. Thus the operation is not shift invariant. (This discussion is related to the FAMlet transform**Y**, i.e., frequency warping of the time domain signal)._{ } __Unitary axes warping,__? In fact after the operation the new function is still on the same axis, the axis is not warped or changed in any way. What happens is that the amplitude values of the function are weighted to compensate for the change in its density and the function is locally stretched (dv/df<1) and shrinked (dv/df>1) according to the actual inner function v. Should we then leave the emphasis of the__U____axis__aside and just use the concept of__unitary warping__?- The operator
**FUF**could also be called a^{-1 }__unitary axis change operator.__At least in the discrete case this operation not only changes the axis but also changes the__resolution__of the representation._{ } - Due to the folding effect many "ideal" operators of the continuous time and frequency case turn "nonideal" when discrete time and frequency are used. For example, the FAM basis is complete only in the continuous case. In the discrete case the base can create a complete picture over a limited range only. This may be even a desired property. Often a compression of the information is needed and it is desired that the signals (functions) are represented with low accuracy and resolution in some areas. Auditory spectrum is a good example of this type of spectral data compression. Due to the folding the unitariness of the warping may also be lost.
- ...

The basic studies related to frequency warping, FAM and FAMlet classes are mainly financed by the Academy of Finland.

The author is grateful to Lic. Tech. Toomas Altosaar for his valuable comments on this text..

Last modified: (©30.10.98/ukl) [ The first version 14.10.98. ]