Spectral leakage and windowing

Transcription

1 EEL33: Discrete-Time Sigals ad Systems Spectral leakage ad widowig. Itroductio Spectral leakage ad widowig I these otes, we itroduce the idea of widowig for reducig the effects of spectral leakage, ad illustrated this cocept with a couple of simple examples.. Spectral leakage ad widowig A. Itroductio Previously, we saw that the DFT samples the DTFT at discrete frequecies; we also observed that whe a sigal cosists oly of frequecies that are iteger multiples of f s N, where N is the legth of the time-domai sequece x [ ] (ad cosequetly the legth of the DFT Xk ( )), those frequecies have ozero coefficiets, while all other coefficiets are zero (see Figures, 3 ad 6 i the Discrete Fourier Trasform otes). However, whe a sigal cosists of frequecies that are ot iteger multiples of f s N, the DFT o loger samples the DTFT at its peaks ad zero crossigs, ad therefore, the DFT coefficiets are o loger zero for frequecies ot i the origial ifiite-legth sigal (see Figures, ad 7 i the Discrete Fourier Trasform otes). I fact, the frequecy cotet is spread out (i.e. ozero) over the full rage of DFT coefficiets Xk ( ). As i the case of the DTFT, this spectral leakage is caused by fiite-legth samplig that occurs for ay practical applicatio. Icreasig the samplig frequecy, thereby geeratig loger discrete-time sequeces for equivalet samplig times, reduces spectral leakage, but does ot elimiate the problem. Oe popular method for mitigatig spectral leakage i the DFT estimatio of spectral cotet is called data widowig. I this method, the origial sigal x [ ], {,,, N }, is modified by multiplicatio with a widowig fuctio that approaches zero ear = ad = N, ad reaches a peak of oe ear = N. While there are may possible choices of widowig fuctios, oe popular choice is the Hammig widow fuctio h [ ], which is give below: π h [ ] =..6 cos ,. () N {,,, N } The role of data widowig is to reduce the artificial high frequecies itroduced i the DFT by fiite-legth samplig. This is perhaps best see through a couple of examples. B. Example # Here, we cosider the sampled sequece x [ ] give by, x [ ] = x c ( f s ) < N elsewhere () where, x c () t = cos[ π( + t], (3) f s = Hz, ad N = 6 ; that is, x c () t is sampled for 3 secods at Hz. Figure below plots x [ ], h [ ] ad x [ ] h [ ], where the operator idicates elemet by elemet multiplicatio of the vectors x [ ] ad h [ ]. Each oe of these sequeces is purposefully plotted for a rage of outside the rage {,,, N }, to illustrate the effect of samplig for oly a fiite legth of time. Coceptually, observe that the fiite-legth samplig process implicitly assumes the sigal x [ ] is zero for values of < ad N; this is equivalet to multiplyig the origial cotiuous-time sigal x c () t by a pulse of width three. This implicit time limitig causes the crisp frequecy cotet of the origial cosie fuctio (at ±( + Hz) to leak to other frequecies throughout the frequecy spectrum. Observe for example, the sharp trasitio from zero to ozero at = ad = N. The frequecy cotet of such a trasitio, while ot preset i the origial cotiuous-time sigal, is a part of the fiite-legth sampled sequece. - -

2 EEL33: Discrete-Time Sigals ad Systems Spectral leakage ad widowig x [ ] h [ ] x [ ] h [ ] Note, however, that whe we multiply the sampled sequece x [ ] by the Hammig widow fuctio h [ ], the sharp trasitios at = ad = N are substatially reduced. I other words, we have replaced multiplicatio by a square pulse with a smoother, tapered pulse istead (i.e. the Hammig widow). Let us ow compare the DTFT ad DFT for the origial sampled sequece x [ ] ad the widowed sampled sequece x [ ] h [ ]. I Figure, we plot the magitude DTFT ad DFT, for the sequeces x [ ] ad x [ ] h [ ]. Note that the DTFT ad DFT for the widowed sequece exhibits sigificatly less spectral leakage tha the origial sigal; specifically, spectral leakage is ow cofied to frequecies aroud the domiat frequecies of the origial cotiuous-time sigal. C. Example # Figure Here, we cosider a slightly more complex sampled sequece x [ ] give by, x [ ] = x c ( f s ) < N elsewhere () - -

3 EEL33: Discrete-Time Sigals ad Systems Spectral leakage ad widowig Mag. DTFT for x [ ] Mag. DFT for x [ ] f - - f Mag. DTFT for x [ ] h [ ] Mag. DFT for x [ ] h [ ] where,. x c () t = + cos[ π( + t] + cos[ π( + t], () f s = Hz, ad N = 6 ; that is, x c () t is agai sampled for 3 secods at Hz. Figure 3 below plots x [ ], h [ ] ad x [ ] h [ ]. As before, each oe of these sequeces is purposefully plotted for a rage of outside the rage {,,, N }, to illustrate the effect of samplig for oly a fiite legth of time. Agai ote that the implicit time limitig of fiite-legth samplig causes the crisp frequecy cotet of the origial cotiuous-time fuctio (at Hz, ± ( + Hz ad ±( + Hz) to leak to other frequecies throughout the frequecy spectrum. Let us ow compare the DTFT ad DFT for the origial sampled sequece x [ ] ad the widowed sampled sequece x [ ] h [ ]. I Figure, we plot the magitude DTFT ad DFT, for the sequeces x [ ] ad x [ ] h [ ]. Note that the DTFT ad DFT for the widowed sequece exhibits sigificatly less spectral leakage tha the origial sigal; specifically, spectral leakage is ow cofied to frequecies aroud the domiat frequecies of the origial cotiuous-time sigal. The discussio above is certaily ot comprehesive i its discussio of widowig methods; it is simply iteded to itroduce basic cocepts i widowig for spectrum estimatio. 3. Coclusio f - - f The Mathematica otebook spectral_leakage.b was used to geerate the examples i this set of otes. I subsequet otes, we will look at how some of the ideas i these ad previous otes ca be applied to the frequecy aalysis of sigals where the frequecy cotet varies over time (e.g. music, speech, etc.). Figure

4 EEL33: Discrete-Time Sigals ad Systems Spectral leakage ad widowig 6 x [ ] h [ ] x [ ] h [ ] Figure 3 - -

5 EEL33: Discrete-Time Sigals ad Systems Spectral leakage ad widowig Mag. DTFT for x [ ] Mag. DFT for x [ ] f f Mag. DTFT for x [ ] h [ ] 6 Mag. DFT for x [ ] h [ ] f - - f Figure - -