# Fourier Transforms for Continuous/Discrete Time/Frequency

The Fourier transform can be defined for signals which are

- discrete or continuous in time, and
- finite or infinite in duration.

- discrete or continuous in frequency, and
- finite or infinite in bandwidth.

Reference [264] develops the DFT in detail--the discrete-time, discrete-frequency case. In the DFT, both the time and frequency axes are finite in length.

Table 2.1 (next page) summarizes the four Fourier-transform cases corresponding to discrete or continuous time and/or frequency.

In all four cases, the Fourier transform can be interpreted as
the *inner product* of the signal
with a complex sinusoid at
radian frequency
[264]:

(3.1) |

where is appropriately adapted,

*e.g.*,

In spectral modeling of audio, we usually deal with indefinitely long
signals. Fourier analysis of an indefinitely long discrete-time
signal is carried out using the Discrete Time Fourier Transform
(DTFT).^{3.1}Below, the DTFT is defined, and selected Fourier theorems are stated
and proved for the DTFT case. Additionally, for completeness, the
Fourier Transform (FT) is defined, and selected FT theorems are stated
and proved as well. Theorems for the DFT case are detailed in
[264].^{3.2}

## Discrete Time Fourier Transform (DTFT)

The *Discrete Time Fourier Transform* (DTFT) can be viewed as the
limiting form of the DFT when its length
is allowed to approach
infinity:

(3.2) |

where denotes the

*continuous*radian frequency variable,

^{3.3}and is the signal amplitude at sample number .

The inverse DTFT is

(3.3) |

which can be derived in a manner analogous to the derivation of the inverse DFT [264].

Instead of operating on sampled signals of length (like the DFT), the DTFT operates on sampled signals defined over all integers .

Unlike the DFT, the DTFT frequencies form a *continuum*. That
is, the DTFT is a function of *continuous* frequency
, while the DFT is a function of discrete
frequency
,
. The DFT frequencies
,
, are given by the angles of
points
uniformly distributed along the unit circle in the complex
plane. Thus, as
, a continuous
frequency axis must result in the limit along the unit circle. The
axis is still finite in length, however, because the time domain
remains sampled.

## Fourier Transform (FT) and Inverse

The *Fourier transform* of a signal
,
, is defined as

and its inverse is given by

Thus, the Fourier transform is defined for continuous time and continuous frequency, both unbounded. As a result, mathematical questions such as existence and invertibility are most difficult for this case. In fact, such questions fueled decades of confusion in the history of harmonic analysis (see Appendix G).

### Existence of the Fourier Transform

Conditions for the *existence* of the Fourier transform are
complicated to state in general [36], but it is *sufficient*
for
to be *absolutely integrable*, *i.e.*,

(3.6) |

This requirement can be stated as , meaning that belongs to the set of all signals having a finite norm ( ). It is similarly sufficient for to be

*square integrable*,

*i.e.*,

(3.7) |

or, . More generally, it suffices to show for [36, p. 47].

There is never a question of existence, of course, for Fourier
transforms of real-world signals encountered in practice. However,
*idealized* signals, such as sinusoids that go on forever in
time, do pose normalization difficulties. In practical engineering
analysis, these difficulties are resolved using Dirac's ``generalized
functions'' such as the *impulse* (also loosely called the
*delta function*), discussed in §B.10.

## Fourier Theorems for the DTFT

This section states and proves selected Fourier theorems for the DTFT.
A more complete list for the DFT case is given in
[264].^{3.4}Since this material was originally part of an appendix, it is
relatively dry reading. Feel free to skip to the next chapter and
refer back as desired when a theorem is invoked.

As introduced in §2.1 above, the Discrete-Time Fourier Transform (DTFT) may be defined as

(3.8) |

We say that is the

*spectrum*of .

### Linearity of the DTFT

(3.9) |

or

(3.10) |

where are any

*scalars*(real or complex numbers), and are any two

*discrete-time signals*(real- or complex-valued functions of the integers), and are their corresponding continuous-frequency

*spectra*defined over the unit circle in the complex plane.

*Proof: *
We have

One way to describe the linearity property is to observe that the Fourier transform ``commutes with mixing.''

### Time Reversal

For any complex signal , , we have

(3.11) |

where .

*Proof: *

Arguably, should include complex conjugation. Let

(3.12) |

denote such a definition. Then in this case we have

(3.13) |

*Proof: *

(3.14) |

In the typical special case of real signals ( ), we have so that

(3.15) |

That is,

*time-reversing a real signal conjugates its spectrum*.

### Symmetry of the DTFT for Real Signals

Most (if not all) of the signals we deal with in practice are real signals. Here we note some spectral symmetries associated with real signals.

#### DTFT of Real Signals

The previous section established that the spectrum of every real signal satisfies

(3.16) |

*I.e.*,

(3.17) |

In other terms, if a signal is real, then its spectrum is

*Hermitian*(``conjugate symmetric''). Hermitian spectra have the following equivalent characterizations:

- The real part is even, while the imaginary part is odd:

- The magnitude is even, while the phase is odd:

#### Real Even (or Odd) Signals

If a signal is *even* in addition to being real, then its DTFT is
also real and even. This follows immediately from the Hermitian
symmetry of real signals, and the fact that the DTFT of any even
signal is real:

This is true since cosine is even, sine is odd, even times even is
even, even times odd is odd, and the sum over all samples of an odd
signal is zero. *I.e.*,

and

If is real and even, the following are true:

Similarly, if a signal is *odd* and real, then its DTFT is odd
and *purely imaginary*. This follows from Hermitian symmetry for
real signals, and the fact that the DTFT of any odd signal is imaginary.

where we used the fact that

and

### Shift Theorem for the DTFT

We define the *shift operator* for sampled signals
by

(3.18) |

where is any integer ( ). Thus, is a

*right-shift*or

*delay*by samples.

The *shift theorem* states^{3.5}

(3.19) |

or, in operator notation,

(3.20) |

*Proof: *

Note that
is a *linear phase term*, so called
because it is a linear function of frequency with slope equal to
:

(3.21) |

The shift theorem gives us that multiplying a spectrum by a linear phase term corresponds to a

*delay*in the time domain by samples. If , it is called a time

*advance*by samples.

### Convolution Theorem for the DTFT

The *convolution* of discrete-time signals
and
is defined
as

(3.22) |

This is sometimes called

*acyclic convolution*to distinguish it from the

*cyclic convolution*used for length sequences in the context of the DFT [264]. Convolution is cyclic in the time domain for the DFT and FS cases (

*i.e.*, whenever the time domain has a finite length), and acyclic for the DTFT and FT cases.

^{3.6}

The *convolution theorem* is then

(3.23) |

That is,

*convolution in the time domain corresponds to pointwise multiplication in the frequency domain*.

*Proof: *The result follows immediately from interchanging the order
of summations associated with the convolution and DTFT:

### Correlation Theorem for the DTFT

We define the *correlation* of discrete-time signals
and
by

The

*correlation theorem*for DTFTs is then

*Proof: *

where the last step follows from the convolution theorem of §2.3.5 and the symmetry result of §2.3.2.

### Autocorrelation

The *autocorrelation* of a signal
is simply the
cross-correlation of
with itself:

(3.24) |

From the

*correlation theorem*, we have

Note that this definition of autocorrelation is appropriate for signals having finite support (nonzero over a finite number of samples). For infinite-energy (but finite-power) signals, such as stationary noise processes, we define the

*sample autocorrelation*to include a normalization suitable for this case (see Chapter 6 and Appendix C).

From the autocorrelation theorem we have that a digital-filter
impulse-response
is that of a *lossless allpass filter*
[263] if and only if
.
In other words, the autocorrelation of the impulse-response of every
allpass filter is impulsive.

### Power Theorem for the DTFT

The *inner product* of two *signals* may be defined in the
time domain by [264]

(3.25) |

The inner product of two

*spectra*may be defined as

(3.26) |

Note that the frequency-domain inner product includes a normalization factor while the time-domain definition does not.

Using inner-product notation, the *power theorem* (or
*Parseval's theorem* [202]) for DTFTs can
be stated as follows:

(3.27) |

That is, the inner product of two signals in the time domain equals the inner product of their respective spectra (a complex scalar in general).

When we consider the inner product of a signal with itself, we have
the special case known as the *energy theorem* (or *Rayleigh's energy theorem*):

(3.28) |

where denotes the norm induced by the inner product. It is always real.

*Proof: *

### Stretch Operator

We define the *stretch operator* in the *time domain* by

(3.29) |

In other terms, we stretch a sampled signal by the factor by inserting

*zeros*in between each pair of samples of the signal.

In the literature on multirate filter banks (see Chapter 11), the
stretch operator is typically called instead the *upsampling*
operator. That is, stretching a signal by the factor of
is called
upsampling the signal by the factor
. (See §11.1.1 for
the graphical symbol (
) and associated discussion.) The
term ``stretch'' is preferred in this book because ``upsampling''
is easily confused with ``increasing the sampling rate''; resampling a
signal to a higher sampling rate is conceptually implemented by a
stretch operation followed by an ideal lowpass filter which moves the
inserted zeros to their properly interpolated values.

Note that we could also call the stretch operator the *scaling*
operator, to unify the terminology in the discrete-time case with that
of the continuous-time case (§2.4.1 below).

### Repeat (Scaling) Operator

We define the *repeat operator* in the *frequency domain* as
a *scaling* of frequency axis by some integer factor
:

(3.30) |

where denotes the radian frequency variable after applying the repeat operator.

The repeat operator maps the entire unit circle (taken as to ) to a segment of itself , centered about , and repeated times. This is illustrated in Fig.2.2 for .

Since the frequency axis is continuous and -periodic for DTFTs, the repeat operator is precisely equivalent to a scaling operator for the Fourier transform case (§B.4). We call it ``repeat'' rather than ``scale'' because we are restricting the scale factor to positive integers, and because the name ``repeat'' describes more vividly what happens to a periodic spectrum that is compressively frequency-scaled over the unit circle by an integer factor.

### Stretch/Repeat (Scaling) Theorem

Using these definitions, we can compactly state the *stretch
theorem*:

(3.31) |

*Proof: *

As traverses the interval , traverses the unit circle times, thus implementing the repeat operation on the unit circle. Note also that when , we have , so that dc always maps to dc. At half the sampling rate , on the other hand, after the mapping, we may have either ( odd), or ( even), where .

The stretch theorem makes it clear how to do
*ideal sampling-rate conversion* for integer upsampling ratios
:
We first stretch the signal by the factor
(introducing
zeros
between each pair of samples), followed by an *ideal lowpass
filter* cutting off at
. That is, the filter has a gain of 1
for
, and a gain of 0 for
. Such a system (if it were realizable) implements *ideal bandlimited interpolation* of the original signal by the factor
.

The stretch theorem is analogous to the *scaling theorem* for
continuous Fourier transforms (introduced in §2.4.1
below).

### Downsampling and Aliasing

The *downsampling* operator
selects every
sample of a signal:

(3.32) |

The *aliasing theorem* states that downsampling in time
corresponds to *aliasing* in the frequency domain:

(3.33) |

where the operator is defined as

(3.34) |

for . The summation terms for are called

*aliasing components*.

In *z* transform notation, the
operator can be expressed as
[287]

(3.35) |

where is a common notation for the primitive th root of unity. On the unit circle of the plane, this becomes

(3.36) |

The frequency scaling corresponds to having a sampling interval of after downsampling, which corresponds to the interval prior to downsampling.

The aliasing theorem makes it clear that, in order to downsample by factor without aliasing, we must first lowpass-filter the spectrum to . This filtering (when ideal) zeroes out the spectral regions which alias upon downsampling.

Note that any rational sampling-rate conversion factor
may be implemented as an upsampling by the factor
followed by
downsampling by the factor
[50,287].
Conceptually, a stretch-by-
is followed by a lowpass filter cutting
off at
, followed by
downsample-by-
, *i.e.*,

(3.37) |

In practice, there are more efficient algorithms for sampling-rate conversion [270,135,78] based on a more direct approach to

*bandlimited interpolation*.

#### Proof of Aliasing Theorem

To show:

or

From the DFT case [264], we know this is true when and are each complex sequences of length , in which case and are length . Thus,

(3.38) |

where we have chosen to keep frequency samples in terms of the original frequency axis prior to downsampling,

*i.e.*, for both and . This choice allows us to easily take the limit as by simply replacing by :

(3.39) |

Replacing by and converting to -transform notation instead of Fourier transform notation , with , yields the final result.

### Differentiation Theorem Dual

**Theorem: **Let
denote a signal with DTFT
, and let

(3.40) |

denote the derivative of with respect to . Then we have

where denotes the DTFT of .

*Proof: *
Using integration by parts, we obtain

An alternate method of proof is given in §B.3.

**Corollary:** Perhaps a cleaner statement is as follows:

This completes our coverage of selected DTFT theorems. The next section adds some especially useful FT theorems having no precise counterpart in the DTFT (discrete-time) case.

## Continuous-Time Fourier Theorems

Selected Fourier theorems for the continuous-time case are stated and proved in Appendix B. However, two are sufficiently important that we state them here.

###

Scaling Theorem

The *scaling theorem* (or *similarity theorem*) says that if
you horizontally ``stretch'' a signal by the factor
in the
time domain, you ``squeeze'' and amplify its Fourier transform by the
same factor in the frequency domain. This is an important general
Fourier duality relationship:

**Theorem: **For all continuous-time functions
possessing a Fourier
transform,

where

and is any nonzero real number (the abscissa stretch factor). A more commonly used notation is the following:

(3.41) |

*Proof: *See §B.4.

The scaling theorem is fundamentally restricted to the continuous-time, continuous-frequency (Fourier transform) case. The closest we come to the scaling theorem among the DTFT theorems (§2.3) is the stretch (repeat) theorem (page ). For this and other continuous-time Fourier theorems, see Appendix B.

### Spectral Roll-Off

**Definition: **A function
is said to be of order
if
there exist
and some positive constant
such
that
for all
.

**Theorem: **(*Riemann Lemma*):
If the derivatives up to order
of a function
exist and are
of bounded variation, then its Fourier Transform
is
asymptotically of order
, *i.e.*,

(3.42) |

*Proof: *See §B.18.

## Spectral Interpolation

The need for spectral interpolation comes up in many situations. For
example, we always use the DFT in practice, while conceptually we
often prefer the DTFT. For *time-limited signals*, that is,
signals which are zero outside some finite range, the DTFT can be
computed from the DFT via *spectral interpolation*. Conversely,
the DTFT of a time-limited signal can be *sampled* to obtain its
DFT.^{3.7}Another application of DFT interpolation is *spectral peak
estimation*, which we take up in Chapter 5; in this
situation, we start with a *sampled* spectral peak from a DFT,
and we use interpolation to estimate the frequency of the peak more
accurately than what we get by rounding to the nearest DFT bin
frequency.

The following sections describe the theoretical and practical details of ideal spectral interpolation.

### Ideal Spectral Interpolation

Ideally, the spectrum of any signal
at any frequency
is obtained by *projecting* the signal
onto the
zero-phase, unit-amplitude, complex sinusoid at frequency
[264]:

(3.43) |

where

Thus, for signals in the DTFT domain which are time limited to , we obtain

(3.44) |

This can be thought of as a zero-centered DFT evaluated at instead of for some . It arises naturally from taking the DTFT of a finite-length signal. Such time-limited signals may be said to have ``finite support'' [175].

### Interpolating a DFT

Starting with a sampled spectrum
,
,
typically obtained from a DFT, we can interpolate by taking the DTFT
of the IDFT which is *not* periodically extended, but instead
*zero-padded* [264]:^{3.8}

(The aliased sinc function,
, is derived in
§3.1.)
Thus, zero-padding in the time domain interpolates a spectrum
consisting of
samples around the unit circle by means of ``
interpolation.'' This is *ideal*,
*time-limited interpolation*
in the frequency domain using the
aliased sinc function as an *interpolation kernel*. We can almost
rewrite the last line above as
,
but such an expression would normally be defined only for
, where
is some integer, since
is
discrete while
is continuous.

Figure F.1 lists a matlab function for performing ideal
spectral interpolation directly in the frequency domain. Such an
approach is normally only used when *non-uniform* sampling of the
frequency axis is needed. For uniform spectral upsampling, it is more
typical to take an inverse FFT, zero pad, then a longer FFT, as
discussed further in the next section.

### Zero Padding in the Time Domain

Unlike time-domain interpolation [270], ideal
spectral interpolation is very easy to implement in practice by means of
*zero padding* in the time domain.
That is,

Since the frequency axis (the unit circle in the plane) is finite in length, ideal interpolation can be implemented

*exactly*to within numerical round-off error. This is quite different from ideal (band-limited)

*time*-domain interpolation, in which the interpolation kernel is sinc ; the sinc function extends to plus and minus infinity in time, so it can never be implemented exactly in practice.

^{3.9}

#### Practical Zero Padding

To interpolate a uniformly sampled spectrum , by the factor , we may take the length inverse DFT, append zeros to the time-domain data, and take a length DFT. If is a power of two, then so is and we can use a Cooley-Tukey FFT for both steps (which is very fast):

(3.45) |

This operation creates new bins between each pair of original bins in , thus increasing the number of spectral samples around the unit circle from to . An example for is shown in Fig.2.4 (compare to Fig.2.3).

In matlab, we can specify zero-padding by simply providing the optional FFT-size argument:

X = fft(x,N); % FFT size N > length(x)

#### Zero-Padding to the Next Higher Power of 2

Another reason we zero-pad is to be able to use a Cooley-Tukey FFT with any
window length
. When
is not a power of
, we append enough
zeros to make the FFT size
be a power of
. In Matlab and
Octave, the function `nextpow2` returns the next higher power
of 2 greater than or equal to its argument:

N = 2^nextpow2(M); % smallest M-compatible FFT size

#### Zero-Padding for Interpolating Spectral Displays

Suppose we perform spectrum analysis on some sinusoid using a length
window. Without zero padding, the DFT length is
. We may
regard the DFT as a *critically sampled DTFT* (sampled in
frequency). Since the bin separation in a length-
DFT is
,
and the zero-crossing interval for Blackman-Harris side lobes is
, we see that there is *one bin per side lobe* in the
sampled window transform. These spectral samples are illustrated for
a Hamming window transform in Fig.2.3b. Since
in
Table 5.2, the main lobe is 4 samples wide when critically
sampled. The side lobes are one sample wide, and the samples happen
to hit near some of the side-lobe zero-crossings, which could be
misleading to the untrained eye if only the samples were shown. (Note
that the plot is clipped at -60 dB.)

If we now *zero pad* the Hamming-window by a factor of 2
(append 21 zeros to the length
window and take an
point
DFT), we obtain the result shown in Fig.2.4. In this case,
the main lobe is 8 samples wide, and there are two samples per side
lobe. This is significantly better for display even though there is
*no new information* in the spectrum relative to Fig.2.3.^{3.10}

Incidentally, the solid lines in Fig.2.3b and 2.4b indicating the ``true'' DTFT were computed using a zero-padding factor of , and they were virtually indistinguishable visually from . ( is not enough.)

#### Zero-Padding for Interpolating Spectral Peaks

For sinusoidal peak-finding, spectral interpolation via zero-padding gets us closer to the true maximum of the main lobe when we simply take the maximum-magnitude FFT-bin as our estimate.

The examples in Fig.2.5 show how zero-padding helps in clarifying the true peak of the sampled window transform. With enough zero-padding, even very simple interpolation methods, such as quadratic polynomial interpolation, will give accurate peak estimates.

Another illustration of zero-padding appears in Section 8.1.3 of [264].

### Zero-Phase Zero Padding

The previous zero-padding example used the *causal* Hamming
window, and the appended zeros all went to the *right* of the
window in the FFT input buffer (see Fig.2.4a). When using
*zero-phase* FFT windows (usually the best choice), the zero-padding
goes in the *middle* of the FFT buffer, as we now illustrate.

We look at zero-phase zero-padding using a *Blackman window*
(§3.3.1) which has good, though
suboptimal, characteristics for audio work.^{3.11}

Figure 2.6a shows a windowed segment of some sinusoidal data, with the window also shown as an envelope. Figure 2.6b shows the same data loaded into an FFT input buffer with a factor of 2 zero-phase zero padding. Note that all time is ``modulo '' for a length FFT. As a result, negative times map to in the FFT input buffer.

Figure 2.7a shows the result of performing an FFT on the data of Fig.2.6b. Since frequency indices are also modulo , the negative-frequency bins appear in the right half of the buffer. Figure 2.6b shows the same data ``rotated'' so that bin number is in order of physical frequency from to . If is the bin number, then the frequency in Hz is given by , where denotes the sampling rate and is the FFT size.

The Matlab script for creating Figures 2.6 and 2.7 is listed in in §F.1.1.

####
Matlab/Octave `fftshift` utility

Matlab and Octave have a simple utility called `fftshift` that
performs this bin rotation. Consider the following example:

octave:4> fftshift([1 2 3 4]) ans = 3 4 1 2 octave:5>If the vector

`[`1 2 3 4] is the output of a length 4 FFT, then the first element (1) is the dc term, and the third element (3) is the point at half the sampling rate ( ), which can be taken to be either plus or minus since they are the same point on the unit circle in the plane. Elements 2 and 4 are plus and minus , respectively. After

`fftshift`, element (3) is first, which indicates that both Matlab and Octave regard the spectral sample at half the sampling rate as a negative frequency. The next element is 4, corresponding to frequency , followed by dc and .

Another reasonable result would be `fftshift([1 2 3 4]) == [4 1
2 3]`, which defines half the sampling rate as a positive frequency.
However, giving
to the negative frequencies balances giving dc
to the positive frequencies, and the number of samples on both sides
is then the same. For an odd-length DFT, there is no point at
, so the result

octave:4> fftshift([1 2 3]) ans = 3 1 2 octave:5>is the only reasonable answer, corresponding to frequencies , respectively.

#### Index Ranges for Zero-Phase Zero-Padding

Having looked at zero-phase zero-padding ``pictorially'' in matlab
buffers, let's now specify the index-ranges mathematically. Denote
the window length by
(an odd integer) and the FFT length by
(a power of 2). Then the windowed data will occupy indices 0
to
(positive-time segment), and
to
(negative-time segment). Here we are assuming a 0-based indexing
scheme as used in C or C++. We add 1 to all indices for matlab
indexing to obtain `1:(M-1)/2+1` and `N-(M-1)/2+1:N`,
respectively. The zero-padding zeros go in between these ranges,
*i.e.*, from
to
.

#### Summary

To summarize, zero-padding is used for

- padding out to the next higher power of 2 so a Cooley-Tukey FFT can be used with any window length,
- improving the quality of spectral displays, and
- oversampling spectral peaks so that some simple final interpolation will be accurate.

*spectral modifications*in the short-time Fourier transform (STFT). This is because spectral modifications cause the time-domain signal to

*lengthen in time*, and without sufficient zero-padding to accommodate it, there will be

*time aliasing*in the reconstruction of the signal from the modified FFTs.

Some examples of interpolated spectral display by means of zero-padding may be seen in §3.4.

**Next Section:**

Spectrum Analysis Windows

**Previous Section:**

Introduction and Overview