## 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:**

The Rectangular Window

**Previous Section:**

Continuous-Time Fourier Theorems