## Effect of Windowing

Let's look at a simple example of windowing to demonstrate what happens when we turn an infinite-duration signal into a finite-duration signal through windowing.

We begin with a sampled complex sinusoid:

(6.14) |

A portion of the real part, , is plotted in Fig.5.3. The imaginary part, , is of course identical but for a 90-degree phase-shift to the right.

The Fourier transform of this infinite-duration signal is a delta
function at
. *I.e.*,
, as indicated in Fig.5.4.

The windowed signal is

(6.15) |

as shown in Fig.5.5. (Note carefully the difference between and .)

The convolution theorem (§2.3.5) tells us that our multiplication in the time domain results in a convolution in the frequency domain. Hence, we will obtain the convolution of with the Fourier transform of the window . This is easy since the delta function is the identity element under convolution ( ). However, since our delta function is at frequency , the convolution shifts the window transform out to that frequency:

(6.16) |

This is shown in Fig.5.6.

From comparing Fig.5.6 with the ideal sinusoidal spectrum in Fig.5.4 (an impulse at frequency ), we can make some observations:

- Windowing in the time domain resulted in a
*``smearing''*or*``smoothing''*in the frequency domain. In particular, the infinitely thin delta function has been replaced by the ``main lobe'' of the window transform. We need to be aware of this if we are trying to resolve sinusoids which are close together in frequency. - Windowing also introduced
*side lobes*. This is important when we are trying to resolve low amplitude sinusoids in the presence of higher amplitude signals. - A sinusoid at amplitude
, frequency
, and phase
manifests (in practical spectrum analysis) as a
*window transform*shifted out to frequency , and scaled by .

As a result of the last point above, the *ideal window transform*
is an *impulse in the frequency domain*. Since this cannot be
achieved in practice, we try to find spectrum-analysis windows which
*approximate* this ideal in some optimal sense. In particular,
we want side-lobes that are as close to zero as possible, and we want
the main lobe to be as tall and narrow as possible. (Since absolute
scalings are normally arbitrary in signal processing, ``tall'' can be
defined as the ratio of main-lobe amplitude to side-lobe
amplitude--or main-lobe energy to side-lobe energy, etc.) There are
many alternative formulations for ``approximating an impulse'', and
each such formulation leads to a particular spectrum-analysis window
which is optimal in that sense. In addition to these windows, there
are many more which arise in other applications. Many commonly used
window types are summarized in Chapter 3.

### Frequency Resolution

The *frequency resolution* of a spectrum-analysis window is
determined by its *main-lobe width* (Chapter 3) in the frequency domain,
where a typical main lobe is illustrated in Fig.5.6
(top). For maximum frequency resolution, we desire the narrowest
possible main-lobe width, which calls for the *rectangular
window* (§3.1), the transform of which is shown in
Fig.3.3. When we cannot be fooled by the large side-lobes
of the rectangular window transform (*e.g.*, when the sinusoids under
analysis are known to be well separated in frequency), the rectangular
window truly is the optimal window for the estimation of frequency,
amplitude, and phase of a sinusoid in the presence of stationary noise
[230,120,121].

The rectangular window has only one parameter (aside from
amplitude)--its *length*. The next section looks at the effect
of an increased window length on our ability to *resolve* two
sinusoids.

#### Two Cosines (``In-Phase'' Case)

Figure 5.7 shows a spectrum analysis of two *cosines*

(6.17) |

where and , and the frequency separation is radians per sample. The zero-padded Fourier analysis uses rectangular windows of lengths , , , and ( , where ). The length FFT output is divided by so that the ideal height of each spectral peak is .

The longest window ( ) resolves the sinusoids very well, while the shortest case ( ) does not resolve them at all (only one ``lump'' appears in the spectrum analysis). In difference-frequency cycles, the analysis windows are two cycles and half a cycle in these cases, respectively. It can be debated whether or not the other two cases are resolved, and we will return to them shortly.

#### One Sine and One Cosine ``Phase Quadrature'' Case

Figure 5.8 shows a similar spectrum analysis of two sinusoids

(6.18) |

using the same frequency separation and window lengths. However, now the sinusoids are 90 degrees out of phase (one sine and one cosine). Curiously, the top-left case ( ) now appears to be resolved! However, closer inspection (see Fig.5.9) reveals that the ``resolved'' spectral peaks are significantly far away from the sinusoidal frequencies. Another curious observation is that the lower-left case ( ) appears worse off than it did in Fig.5.7, and worse than the shorter-window analysis at the top right of Fig.5.8. Only the well resolved case at the lower right (spanning two full cycles of the difference frequency) appears unaffected by the relative phase of the two sinusoids under analysis.

Figure 5.9 shows the same plots as in
Fig.5.8, but overlaid. From this we can see that the peak
locations are *biased* in under-resolved cases, both in amplitude
and frequency.

The preceding figures suggest that, for a rectangular window of length
, two sinusoids are well *resolved* when they are separated in
frequency by

(6.19) |

where the frequency-separation is in radians per sample. In cycles per sample, the inequality becomes

(6.20) |

where the denotes normalized frequency in cycles per sample. In Hz, we have

(6.21) |

or

(6.22) |

Note that is the number of samples in one period of a sinusoid at frequency Hz, sampled at Hz. Therefore, we have derived a rule of thumb for frequency resolution that requires at least

*two full cycles of the difference-frequency*under the rectangular window.

A more detailed study [1] reveals that cycles of the difference-frequency is sufficient to enable fully accurate peak-frequency measurement under the rectangular window by means of finding FFT peaks. In §5.5.2 below, additional minimum duration specifications for resolving closely spaced sinusoids are given for other window types as well.

In principle, we can resolve *arbitrarily small* frequency
separations, provided

- there is no noise, and
- we are sure we are looking at the sum of two ideal sinusoids under the window.

The rectangular window provides an abrupt transition at its edge. While it remains the optimal window for sinusoidal peak estimation, it is by no means optimal in all spectrum analysis and/or signal processing applications involving spectral processing. As discussed in Chapter 3, windows with a more gradual transition to zero have lower side-lobe levels, and this is beneficial for spectral displays and various signal processing applications based on FFT methods. We will encounter such applications in later chapters.

**Next Section:**

Resolving Sinusoids

**Previous Section:**

Spectrum of a Windowed Sinusoid