Spectrum analysis using the discrete fourier transform  (Page 3/3)

Effects of windowing

Applying the DTFT multiplication property $$(X({\omega }_k))=\sum_{n=-\infty }^{\infty } x(n)w(n)e^{-(i{\omega }_{k}n)}=\frac{1}{2\pi }(X({\omega }_k), W({\omega }_k))$$ we find that the DFT of the windowed (truncated) signal produces samples not of the true (desired) DTFT spectrum $X(\omega )$ , but of a smoothed verson $(X(\omega ), W(\omega ))$ . We want this to resemble $X(\omega )$ as closely as possible, so $W(\omega )$ should be as close to an impulse as possible. The window $w(n)$ need not be a simple truncation (or rectangle , or boxcar ) window; other shapes can also be used as long as they limit the sequence to at most $N$ consecutive nonzero samples. All good windows are impulse-like, and represent various tradeoffsbetween three criteria:

• main lobe width: (limits resolution of closely-spaced peaks of equal height)
• height of first sidelobe: (limits ability to see a small peak near a big peak)
• slope of sidelobe drop-off: (limits ability to see small peaks further away from a big peak)

Many different window functions have been developed for truncating and shaping a length- $N$ signal segment for spectral analysis.The simple truncation window has a periodic sinc DTFT, as shown in . It has the narrowest main-lobe width, $\frac{2\pi }{N}$ at the -3 dB level and $\frac{4\pi }{N}$ between the two zeros surrounding the main lobe, of the common window functions, but also the largest side-lobe peak, at about -13 dB.The side-lobes also taper off relatively slowly.

The Hann window (sometimes also called the hanning window), illustrated in , takes the form $w(n)=0.5-0.5\cos \left(\frac{2\pi n}{N-1}\right)$ for $n$ between $0$ and $N-1$ . It has a main-lobe width (about $\frac{3\pi }{N}$ at the -3 dB level and $\frac{8\pi }{N}$ between the two zeros surrounding the main lobe) considerably larger than the rectangular window,but the largest side-lobe peak is much lower, at about -31.5 dB. The side-lobes also taper off much faster.For a given length, this window is worse than the boxcar window at separating closely-spaced spectral components of similar magnitude, but better for identifyingsmaller-magnitude components at a greater distance from the larger components.

The Hamming window , illustrated in , has a form similar to the Hann window but with slightly different constants: $w(n)=0.538-0.462\cos \left(\frac{2\pi n}{N-1}\right)$ for $n$ between $0$ and $N-1$ . Since it is composed of the same Fourier series harmonics as the Hann window,it has a similar main-lobe width (a bit less than $\frac{3\pi }{N}$ at the -3 dB level and $\frac{8\pi }{N}$ between the two zeros surrounding the main lobe), but the largest side-lobe peak is much lower, at about -42.5 dB.However, the side-lobes also taper off much more slowly than with the Hann window. For a given length, the Hamming window is better than the Hann (and of coursethe boxcar) windows at separating a small component relatively near to a large component, but worse than the Hann for identifying very small components atconsiderable frequency separation. Due to their shape and form, the Hann and Hamming windows are also known as raised-cosine windows .

The main-lobe width of a window is an inverse function of the window-length $N$ ; for any type of window, a longer window will always provide better resolution.

Many other windows exist that make various other tradeoffs between main-lobe width, height of largest side-lobe, and side-lobe rolloff rate.The Kaiser window family, based on a modified Bessel function, has an adjustable parameter that allows the user to tune the tradeoff over a continuous range.The Kaiser window has near-optimal time-frequency resolution and is widely used. A list of many different windows can be found here .

shows 64 samples of a real-valued signal composed of several sinusoids of various frequencies and amplitudes.

shows the spectral estimate produced using a length-64 Hamming window applied to the same signal shown in .

shows the spectral estimate produced using a length-64 Hann window applied to the same signal shown in .

This example illustrates that no single window is best for all spectrum analyses. The best window depends on the nature of the signal, and different windows maybe better for different components of the same signal. A skilled spectrum analysist may apply several different windows to a signal togain a fuller understanding of the data.