# Derivation of the Discrete Fourier Transform (DFT)

This chapter derives the Discrete Fourier Transform (DFT) as a projection of a length signal onto the set of sampled complex sinusoids generated by the th roots of unity.

## Geometric Series

Recall that for any complex number , the signal

*geometric sequence*,

*i.e.*, each term is obtained by multiplying the previous term by the (complex) constant . A

*geometric series*is the

*sum*of a geometric sequence:

*closed form*:

*Proof: *We have

When , , by inspection of the definition of .

## Orthogonality of Sinusoids

A key property of sinusoids is that they are *orthogonal at different
frequencies*. That is,

For length *sampled* sinusoidal signal segments, such as used
by the DFT, exact orthogonality holds only for the *harmonics of
the sampling-rate-divided-by-*, *i.e.*, only for the frequencies (in Hz)

*whole number of periods in samples*(depicted in Fig.6.2 for ).

^{6.1}

The complex sinusoids corresponding to the frequencies are

*roots of unity*in the complex plane.

### Nth Roots of Unity

As introduced in §3.12, the complex numbers

*th roots of unity*because each of them satisfies

*primitive th root of unity*.

^{6.2}

The th roots of unity are plotted in the complex plane in Fig.6.1 for . It is easy to find them graphically by dividing the unit circle into equal parts using points, with one point anchored at , as indicated in Fig.6.1. When is even, there will be a point at (corresponding to a sinusoid with frequency at exactly half the sampling rate), while if is odd, there is no point at .

### DFT Sinusoids

The sampled sinusoids generated by integer powers of the roots of
unity are plotted in Fig.6.2. These are the sampled sinusoids
used by the
DFT. Note that taking successively higher integer powers of the
point on the unit circle
*generates* samples of the th DFT sinusoid, giving ,
. The th sinusoid generator is in turn
the th th root of unity (th power of the primitive th root
of unity ).

Note that in Fig.6.2 the range of is taken to be instead of . This is the most ``physical'' choice since it corresponds with our notion of ``negative frequencies.'' However, we may add any integer multiple of to without changing the sinusoid indexed by . In other words, refers to the same sinusoid for all integers .

## Orthogonality of the DFT Sinusoids

We now show mathematically that the DFT sinusoids are exactly orthogonal. Let

where the last step made use of the closed-form expression for the sum of a geometric series (§6.1). If , the denominator is nonzero while the numerator is zero. This proves

## Norm of the DFT Sinusoids

For , we follow the previous derivation to the next-to-last step to get

## An Orthonormal Sinusoidal Set

We can normalize the DFT sinusoids to obtain an orthonormal set:

*normalized DFT sinusoids*. In §6.10 below, we will project signals onto them to obtain the

*normalized DFT*(NDFT).

## The Discrete Fourier Transform (DFT)

Given a signal
, its DFT is defined
by^{6.3}

*spectrum*of , and is the th

*sample*of the spectrum at frequency . Thus, the th sample of the spectrum of is defined as the inner product of with the th DFT sinusoid . This definition is times the

*coefficient of projection*of onto ,

*i.e.*,

In summary, the DFT is proportional to the set of coefficients of projection onto the sinusoidal basis set, and the inverse DFT is the reconstruction of the original signal as a superposition of its sinusoidal projections. This basic ``architecture'' extends to all linear orthogonal transforms, including wavelets, Fourier transforms, Fourier series, the discrete-time Fourier transform (DTFT), and certain short-time Fourier transforms (STFT). See Appendix B for some of these.

We have defined the DFT from a geometric signal theory point of view, building on the preceding chapter. See §7.1.1 for notation and terminology associated with the DFT.

## Frequencies in the ``Cracks''

The DFT is defined only for frequencies . If we are analyzing one or more periods of an exactly periodic signal, where the period is exactly samples (or some integer divisor of ), then these really are the only frequencies present in the signal, and the spectrum is actually zero everywhere but at , . However, we use the DFT to analyze arbitrary signals from nature. What happens when a frequency is present in a signal that is not one of the DFT-sinusoid frequencies ?

To find out, let's project a length segment of a sinusoid at an arbitrary frequency onto the th DFT sinusoid:

The coefficient of projection is proportional to

using the closed-form expression for a geometric series sum once
again. As shown in §6.3-§6.4 above,
the sum is if
and zero at , for . However,
the sum is *nonzero at all other frequencies* .

Since we are only looking at samples, any sinusoidal segment can be
projected onto the DFT sinusoids and be reconstructed exactly by a
linear combination of them. Another way to say this is that the DFT
sinusoids form a *basis* for , so that any length signal
whatsoever can be expressed as a linear combination of them. Therefore, when
analyzing segments of recorded signals, we must interpret what we see
accordingly.

The typical way to think about this in practice is to consider the DFT
operation as a *digital filter* for each , whose input is
and whose output is
at time .^{6.4} The
*frequency response* of this filter is what we just
computed,^{6.5} and its magnitude is

*sidelobes*of the DFT response, while the main peak may be called the

*main lobe*of the response. Since we are normally most interested in spectra from an audio perspective, the same plot is repeated using a

*decibel*vertical scale in Fig.6.3b

^{6.6}(clipped at dB). We see that the sidelobes are really quite high from an audio perspective. Sinusoids with frequencies near , for example, are only attenuated approximately dB in the DFT output .

We see that
is sensitive to *all* frequencies between dc
and the sampling rate *except* the other DFT-sinusoid frequencies
for . This is sometimes called *spectral leakage*
or *cross-talk* in the spectrum analysis. Again, there is *no
leakage* when the signal being analyzed is truly periodic and we can choose
to be exactly a period, or some multiple of a period. Normally,
however, this cannot be easily arranged, and spectral leakage can
be a problem.

Note that peak spectral leakage is not reduced by increasing
.^{6.7} It can be thought of as being caused by abruptly
*truncating* a sinusoid at the beginning and/or end of the
-sample time window. Only the DFT sinusoids are not cut off at the
window boundaries. All other frequencies will suffer some truncation
distortion, and the spectral content of the abrupt cut-off or turn-on
transient can be viewed as the source of the sidelobes. Remember
that, as far as the DFT is concerned, the input signal is the
same as its
*periodic extension* (more about this in
§7.1.2). If we repeat samples of a sinusoid at frequency
(for any
), there will be a ``glitch''
every samples since the signal is not periodic in samples.
This glitch can be considered a source of new energy over the entire
spectrum. See
Fig.8.3 for an example waveform.

To reduce spectral leakage (cross-talk from far-away
frequencies), we typically use a
*window*
function, such as a
``raised cosine'' window, to *taper* the data record gracefully
to zero at both endpoints of the window. As a result of the smooth
tapering, the *main lobe widens* and the *sidelobes
decrease* in the DFT response. Using no window is better viewed as
using a *rectangular window* of length , unless the signal is
exactly periodic in samples. These topics are considered further
in Chapter 8.

## Spectral Bin Numbers

Since the th spectral sample
is properly regarded as
a measure of spectral amplitude over a *range* of frequencies,
nominally to , this range is sometimes called a
*frequency bin*
(as in a ``storage bin'' for spectral energy).
The frequency index is called the *bin number*, and
can be regarded as the total energy in the th
bin (see §7.4.9).
Similar remarks apply to samples of any
bandlimited
function; however, the term ``bin'' is only used in the frequency
domain, even though it could be assigned exactly the same meaning
mathematically in the time domain.

## Fourier Series Special Case

In the very special case of *truly periodic* signals
, for all
, the DFT may be regarded as
computing the *Fourier series coefficients* of from one
period of its sampled representation ,
. The
period of must be exactly seconds for this to work. For the
details, see §B.3.

## Normalized DFT

A more ``theoretically clean'' DFT is obtained by projecting onto the
*normalized DFT sinusoids* (§6.5)

*normalized DFT (NDFT)*of is

It can be said that only the NDFT provides a proper *change of
coordinates* from the time-domain (shifted impulse basis signals) to
the frequency-domain (DFT sinusoid basis signals). That is, only the
NDFT is a pure
*rotation* in , preserving both orthogonality and the unit-norm
property of the basis functions. The DFT, in contrast, preserves
orthogonality, but the norms of the basis functions grow to
. Therefore, in the present context, the DFT coefficients can be
considered ``denormalized'' frequency-domain coordinates.

## The Length 2 DFT

The length DFT is particularly simple, since the basis sinusoids are real:

The DFT sinusoid is a sampled constant signal, while is a sampled sinusoid at half the sampling rate.

Figure 6.4 illustrates the graphical relationships for the length DFT of the signal .

Analytically, we compute the DFT to be

and the corresponding projections onto the DFT sinusoids are

Note the lines of orthogonal projection illustrated in the figure. The ``time domain'' basis consists of the vectors , and the orthogonal projections onto them are simply the coordinate axis projections and . The ``frequency domain'' basis vectors are , and they provide an orthogonal basis set that is rotated degrees relative to the time-domain basis vectors. Projecting orthogonally onto them gives and , respectively. The original signal can be expressed either as the vector sum of its coordinate projections (0,...,x(i),...,0), (a time-domain representation), or as the vector sum of its projections onto the DFT sinusoids (a frequency-domain representation of the time-domain signal ). Computing the coefficients of projection is essentially ``taking the DFT,'' and constructing as the vector sum of its projections onto the DFT sinusoids amounts to ``taking the inverse DFT.''

In summary, the oblique coordinates in Fig.6.4 are interpreted as follows:

## Matrix Formulation of the DFT

The DFT can be formulated as a complex matrix multiply, as we show in this section. (This section can be omitted without affecting what follows.) For basic definitions regarding matrices, see Appendix H.

The DFT consists of inner products of the input signal with sampled complex sinusoidal sections :

*DFT matrix*,

*i.e.*,

The notation
denotes the
*Hermitian transpose* of the complex matrix (transposition
and complex conjugation).

Note that the th column of is the th DFT sinusoid, so that the th row of the DFT matrix is the complex-conjugate of the th DFT sinusoid. Therefore, multiplying the DFT matrix times a signal vector produces a column-vector in which the th element is the inner product of the th DFT sinusoid with , or , as expected.

Computation of the DFT matrix in Matlab is illustrated in §I.4.3.

The *inverse DFT matrix* is simply
. That is,
we can perform the inverse DFT operation as

Since the forward DFT is , substituting from Eq.(6.2) into the forward DFT leads quickly to the conclusion that

This equation succinctly states that the

*columns of are orthogonal*, which, of course, we already knew.

*I.e.*, for , and :

The *normalized DFT matrix* is given by

*inverse*DFT matrix is simply , so that Eq.(6.3) becomes

*orthonormal*. Such a complex matrix is said to be

*unitary*.

When a *real* matrix
satisfies
, then
is said to be
*orthogonal*.
``Unitary'' is the generalization of ``orthogonal'' to
complex matrices.

## DFT Problems

See `http://ccrma.stanford.edu/~jos/mdftp/DFT_Problems.html`

**Next Section:**

Fourier Theorems for the DFT

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Geometric Signal Theory