The time-domain multiplication given in (7.3.3) is equivalent to the convolution in the
frequency domain. Thus the DFT of x
n can be expressed as
k Wk Ã Xk
Wk À lXk,
where W(k) is the DFT of the window function w(n), and X(k) is the true DFT of the signal
x(n). Equation (7.3.4) shows that the computed spectrum consists of the true spectrum
X(k) convoluted with the window function's spectrum W(k). This means that when we
apply a window to a signal, the frequency components of the signal will be corrupted in
the frequency domain by a shifted and scaled version of the window's spectrum.
As discussed in Section 5.3.3, the magnitude response of the rectangular window
defined in (7.3.2) can be expressed as
It consists of a mainlobe of height N at ! 0, and several smaller sidelobes. The
frequency components that lie under the sidelobes represent the sharp transition of
w(n) at the endpoints. The sidelobes are between the zeros of W!, with center
frequencies at !
À 2, . . ., and the first sidelobe is down only 13 dB
from the mainlobe level. As N increases, the height of the mainlobe increases and its
width becomes narrower. However, the peak of the sidelobes also increases. Thus the
ratio of the mainlobe to the first sidelobe remains the same about 13 dB.
The sidelobes introduce spurious peaks into the computed spectrum, or to cancel true
peaks in the original spectrum. The phenomenon is known as spectral leakage. To avoid
spectral leakage, it is necessary to use a shaped window to reduce the sidelobe effects.
Suitable windows have a value of 1at n M, and are tapered to 0 at points n 0 and
n N À 1to smooth out both ends of the input samples to the DFT.
If the signal x(n) consists of a single sinusoid, that is,
the spectrum of the infinite-length sampled signal over the Nyquist interval is given as
À p ! p,
which consists of two line components at frequencies
. However, the spectrum of the
windowed sinusoid defined in (7.3.3) can be obtained as
W! À !
where W! is the spectrum of the window function.
Equation (7.3.8) shows that the windowing process has the effect of smearing the
original sharp spectral line d! À !
at frequency !
and replacing it with W! À !