Sound Analysis
83
bins
leakage
spectral resolution
temporal resolution
uncertainty principle
transition bandwidth
passband
stopband
adjustable windows
transition bandwidth
zero padding
least a hop size equal to R/2 in order to preserve all information at the analysis
stage. Moreover, the larger the main-lobe width, the more difficult is to separate
two frequency components that are close to each other. In other words, we have
a reduction in frequency resolution for windows with a large main lobe.
The side-lobe level indicates how much a sinusoidal component affects the
DFT bins nearby. This phenomenon, called leakage, can induce an analysis
procedure to detect false spectral peaks, or measurements on actual peaks can
be affected by errors. For a given resolution considered to be acceptable, it is
desirable that the side-lobe level be as small as possible.
The window length is chosen according to the tradeoff between spectral
resolution and temporal resolution governed by the uncertainty principle. The
STFT analysis is based on the assumption that, within one frame, the signal is
stationary. The more the window is short, the closer the assumption is to truth,
but short windows determine low spectral resolution.
The windows described in this section have a fixed shape. When they are
multiplied by an ideal lowpass impulse response they impose a fixed transition
bandwidth, i.e. a certain frequency space between the passband and the stop-
band. There are other, more versatile windows, that allow to tune their behavior
by means of a parameter. The most widely used of these adjustable windows
is the Kaiser window [58], whose parameter can be related to the transition
bandwidth.
Zero padding
It is quite common to use a window whose length R is smaller than the number
N of points used to compute the DFT. In thise way, we have a spectrum repre-
sentation on a larger number of points, and the shape of the frequency response
can be understood more easily. Usually, the sequency of R points is extended
by means of N - R zeros, and this operation is called zero padding. Extending
the time response with zeros corresponds to sampling the frequency response
more densely, but it does not introduce any increase in frequency resolution. In
fact, the resolution is only determined by the length and shape of the effective
window, and additional zeros can not change it.
Consider the zero-padded signal
y(n) =
x(n) n = 0, . . . , R - 1
0
n = R, . . . , N - 1
.
(13)
The DFT is found as
Y (k) =
N -1
n=0
y(n)e
-j2kn
N
=
R-1
n=0
y(n)e
-j2kn
N
= Resampling
N
(X, R) ,
(14)
where the notation Resampling
N
(X, R) indicates the resampling on N points
of R points of the discrete-time signal X, obtained as DFT(x) = X.
Exercise
Draw the time-domain shape and the frequency response of each of the windows
of table 4.1. Then, using a Rectangular, a Hann, and a Blackman window,
analyze the signal
x(n) = 0.8 sin (2f
1
n/F
s
) + sin (2f
2
n/F
s
) ,
(15)