Practical Spectral Aaysis (cotiue) (from Boaz Porat s book) Frequecy Measuremet Oe of the most importat applicatios of the DFT is the measuremet of frequecies of periodic sigals (eg., siusoidal sigals), which is essetial to spectral aalysis. It is more coveiet to see the case of comple epoetial sigals first, ad proceed to real-valued siusoids later. I practice, real-valued siusoids are much commo. However, i certai applicatios (such as radar ad commuicatio), comple sigals appear.
The simplest case of a sigal for which frequecy measuremet is a meaigful problem is a sigle comple epoetial: ad we wish to measure the frequecy w 0. We sample the sigal at iterval T ad collect cosecutive data poits, thus gettig the discrete-time sigal where 0 = w 0 T. We assume that w 0 T, so measuremet of 0 implies uambiguous measuremet of w 0.
The DTFT of the sampled sigal is give by where D(.,.) is the Dirchlet kerel. I particular, D(0,)=, evaluatio of DTFT at the frequecy = 0 gives Furthermore, sice D(- 0,,) < for all 0, the poit = 0 is the uique global maimum of Y f () o. We therefore coclude that 0 ca be obtaied by fidig the poit of global maimum of Y f () i the frequecy (,).
To reiterate I practice, it is impossible to fid the global maimum of Y f () eactly, sice we caot evaluate this fuctio at the ifiite umber of frequecy poits. A first approimatio ca be obtaied by computig the DFT of y ad searchig for the poit of maimum of Y d (k), the DFT of y. The ide k 0 for which Y d (k) is maimized yields a correspodig frequecy 2k 0 /. Better approimatio ca be obtaied if ecessary, by either zero paddig or usig a Chirp trasform to icrease the resolutio of samplig i the DTFT domai.
Frequecy measuremet for two comple epoetials Proceedig to a more difficult problem, we ow cosider a sigal cosistig of two comple epoetials: Our aim is to measure the frequecies ω ad ω2. As before, we sample the sigal at iterval T ad collect measuremets of the sampled sigal to obtai
The Fourier trasform of the sampled sigal is give by I particular, evaluatio of the Fourier trasform at the frequecy θ=θ gives We observe the followig:
peaks (local maima) are separated but shifted i (a),(b),(d),(e)
Refiemet by widowig
We ca icrease the chaces of meetig the later coditio. For eample, if we use a Kaiser widow with side-lobe level of 80dB, we will be able to hadle siusoidal compoets whose amplitudes differ by up to four orders of magitude. However, this comes at the price of makig the frequecy separatio coditio more difficult to meet, due to the wideig of the mai lobe.
peaks (local maima) are ot oly separated but also less shifted i (a), (b), (d), (e)
Practice of frequecy measuremet i spectral aalysis Based o the above studies, practical frequecy measuremet usig DFT cosists, as a miimum, of the followig three steps:. Multiplicatio of the sampled sequece by a widow. 2. Computatio of the DFT, usually through FFT. 3. Search for the local maima of the absolute value of the DFT ad selectio of the maima of iterest. We omit the case whe the sigal is corrupted with oises. Refer to Boaz Porat s book for more iformatio. Additioal steps are ecessary if measuremet of the amplitudes ad phases is required as well, but we shall ot discuss them here. As we have eplaied, the choice of a widow requires kowledge of the ature of the sigal.
Review of Decimatio-i-time FFT algorithm A characteristic of DFT is that it ca be computed very fast algorithms: DFT pairs: X k 0 W k0 k, X k W k, k 0,..., 0,..., W = e j2/ is a root of the equatio W =. It requires 2 comple multiplicatios ad () comple additios for direct computatio.
Review of Decimatio-i-time FFT algorithm Most coveietly illustrated by cosiderig the special case of a iteger power of 2, i.e, =2 v. Separatig ito two (/2)-poit sequece cosistig of the eve umbered ad odd-umbered poits i, respectively. X k eve W k odd W k
I sum, ( /2) ( /2) rk k rk /2 /2 r0 r0 X k 2 r( W ) W 2r ( W ) k G k W H k, k 0,,..., Both Gk ad Hk ca be computed by (/2)-poit DFT Gk: the (/2)-poit DFT of the eve umbered poits of the origial sequece H(k): the (/2)-poit DFT of the odd-umbered poit of the origial sequece. Although the ide rages over values, k = 0,,, -, they must be computed oly for k betwee 0 ad (/2)-, sice Gk ad Hk are each periodic i k with period /2.
Aother method: Decimatio-i-frequecy FFT algorithm The above decimatio-i-time FFT algorithms are based o structurig the DFT computatio by formig smaller ad smaller subsequeces of the iput sequece. Alteratively, we ca cosider dividig the output sequece Xk ito smaller ad smaller subsequeces i the same maer. The eve-umbered frequecy samples are So, 0,,..., 0 k W k X k / 2) ( ) (2 2) / ( 0 ) (2 0 ) (2 2 r r r W W W r X 2) / ( 0 / 2)) ( ( 2 / 2) ( 0 2 2) / ( 2 r r W W r X
Sice W 2r ( /2) W 2r W r W 2r ad W 2 W / 2 X2r ( / 2) ( 0 ( / 2)) W r / 2 r 0,,...,( / 2) The above equatio is the (/2)-poit DFT of the (/2)-poit sequece obtaied by addig the first ad the last half of the iput sequece. Aalogical meaig: Addig the two halves of the iput sequece represets time aliasig, cosistet with the fact that i computig oly the eve-umber frequecy samples, we are subsamplig the DFT of.
We ow cosider obtaiig the odd-umbered frequecy poits: Sice 2) / ( ) (2 2) / ( 0 ) (2 0 ) (2 2 r r r W W W r X ) (2 2) / ( 0 ) (2 2) / ( 0 ) 2)( 2 / ( 2) / ( 0 ) 2)( 2 / ( 2 / ) (2 2) / ( 2) / ( 2) / ( r r r r r W W W W W (Because W (/2)(2r+) = ( ) (2r+) = )
We obtai X2r ( / 2) 0 ( ( / 2) 0 ( / 2) W / 2) W W r / 2 (2r) r 0,,...,( / 2) The above equatio is the (/2)-poit DFT of the sequece obtaied by subtractig the secod half of the iput sequece from the first half, ad the multiplyig the resultig sequece by W. Let g = ++/2 ad h = +/2, the DFT ca be computed by formig the sequeces g ad h, the computig hw, ad fially computig the (/2)-poit DFTs of these two sequeces.
Flow graph of decimatio-i-frequecy decompositio of a - poit DFT (=8).
Recursively, we ca further decompose the (/2)-poit DFT ito smaller substructures:
Fially, we have
Butterfly structure for decimatio-i-frequecy FFT algorithm: The decimatio-i-frequecy FFT algorithm also has the computatio compleity of O( log 2 )
Circular Covolutio (for DFT) Time-domai covolutio implies frequecy domai multiplicatio. This property is valid for cotiuous Fourier trasform, Fourier series, ad DTFT, but is ot eactly true for DFT. The DFT pair cosidered hereafter (followig Opeheim s book, where the / is put o the iversetrasform side): X k 0 k0 W k, X k W k, k 0,..., 0,..., where W = e j2/ is a root of the equatio W =.
Circular Covolutio (for DFT) For DFT, time domai circular covolutio implies frequecy domai multiplicatio, ad vice versa. Cosider a periodic sequece. Its DTFT is both periodic ad discrete i frequecy. Multiplicatio i the frequecy domai results i a covolutio of the two correspodig periodic sequeces i the time domai. ow let s cosider a sigle period of the resulted sequece. Sice the two sequeces are both periodic, the covolutio appears as foldig the rear of a sequece to the frot oe by oe, ad superimposig the ier products so obtaied, i a sigle period.
Covolutio of two periodic sequeces
Circular covolutio (defiitio) Symbol for represetig circular covolutio: or. ) mod ( )) (( 0 2 0 2 2 2 3 m m m m m m circular covolutio
Compute the ier product of k ad h-k for each. Recall: Liear covolutio ca be realized by Reflect hk about the origi to obtai h-k. Shift the origi of the reflected sequeces to k= i tur for each.
Liear covolutio of a square wave
Outcome
Imagig circular covolutio as covolutio o a circle
Eample: circular covolutio of 2 m with a delayed impulse sequece m m
Eample: circular covolutio of two rectagular pulses -poit circular covolutio of two sequeces of legth.
Eample: circular covolutio of two rectagular pulses Magitude becomes -poit circular covolutio of two sequeces of legth.
Circular covolutio of two rectagular pulses with zero paddig Give two sequeces of legth L, assume that we add L zeros o its ed, makig a =2L poit sequece referred to as zero paddig -poit circular covolutio of two sequeces of legth L, where =2L.
-poit circular covolutio of two sequeces of legth L, where =2L (cotiue). As ca be see i the figures, by zero paddig, we ca use circular covolutio to compute liear covolutio of two fiite legth sigals.
Circular Covolutio vs. DFT For DFT Time domai circular covolutio implies DFT frequecy domai multiplicatio: Time domai multiplicatio implies DFT frequecy domai circular covolutio (with / amplitude reductio): 2 3 2 3 k X k X k X 2 3 2 3 k X k X k X
Some other properties ivolvig circulatio: Time domai circular shift implies frequecy domai phase shift: (( m)), 0 e j(2k / ) m X k W km X k Duality property of DFT: Sice DFT ad IDFT has very similar form, we have a duality property for DFT: If DFT X k The X DFT (( k)), 0 k
DFT Properties: