be a deterministic function that satisfies x( t) dt. Then its Fourier

Transcription

1 Lecture Fourier ransforms and Applications Definition Let ( t) ; t (, ) be a deterministic function that satisfies ( t) dt hen its Fourier it ransform is defined as X ( ) ( t) e dt ( )( ) heorem he inverse Fourier transform of X ( ) is it ( ) ( ) ( )( ) t X e d X t it i it i( t) Proof: Begin ith X ( ) e d ( ) e d e d ( ) e d d Let s t hen ds d and, so that st i t i s X ( ) e d ( s t) e d ds s We no need to compute the inner integral Rather than using a limit-based argument associated ith the infinite limits of integration, e ill use the folloing method: he Fourier transform of () s is is ( s) e ds Hence, the inverse Fourier transform of is s is e d ( s) But this gives i s e d ( s) Hence: s s it X ( ) e d ( s t) ( s) ds ( t) his completes the proof s Remark As a result of this proof, e have discovered the folloing Fourier transform pairs: ( t) and ( ) Remark he units of are typically radians/second, though radians per hour, day or year are also common Let f / have units of cycles/second (ie Hz) hen the Fourier transform pair becomes: i f t X ( f ) ( t) e dt and he Fourier ransform of the Autocorrelation Function- ( t) X ( f ) e i f t df Eample Consider a Gauss-Markov random process ith R () e hen using calculus, ( i) ( i ) i ( i ) e e i ( R )( ) e e d e e d e e e e ( i) i i

2 Hence, ( R )( f) in units of poer/hz ( f ) Definition For a ss random process ith S ( ) ( R )( ) is called the poer spectral density (psd) for so, its units are in poer per Hz; not poer per rad/sec NOE : he reason for this eirdness of the units stems from the factor included in the inverse Fourier transform It is that factor that changes the units from poer per rad/sec to poer per Hz] / NOE : One might ell ask about the use of the term poer After all, the units of R ( ) E[ ( t) ( t )] is the square of the units of eample, if ( t) / R t () t () he anser is that often the squared units of t () t () are related to the units of poer For is the voltage across a resistor having resistance R, then the poer associated ith this voltage is, hich has units of volts /Ohm If R, then the numerical value of the poer is () t ] Even Eample continued For 5 S ( ) 5, Figure he autocorrelation and psd functions for a GM process ith 5 rad / sec Eample Continuous-time hite noise has S ( ) for all (, ) R here ( ) ( ) ( ) Since the total poer is the area associated ith is the Dirac-delta function Hence, the psd is S ( ), the hite noise process has infinite variance he parameter is called the variance intensity Clearly, such a process does not eist On the other hand, consider band-limited hite noise that has S( ) cfor all (, ) We ill no recover R () for this process i i i i i i e c e e c e e R ( ) ce d i i i Recalling that e i e i i c c sin( ) c sin gives R ( ) sin( ) Sa( ) i c c R () ce d ( ) / We can also rite this as Notice that c c / f / f Hence, for a specified R (), the psd constant is

3 3 Hence, the constant c has units of pr/hz and the to-sided bandidth has units of Hz A plot of for and is shon at right Notice that the first null in 5 Sa( ) f occurs hen, or hen R () Hence, the ider the bandidth, the narroer the autocorrelation function and the larger R () / Figure Plot of R () for 5 From Eample, e have the folloing Fourier transform pair: a for a X t Sa t otherise ( ) ( ) ( ) Eample 3 he Sa( a) function described in Eample is an etremely important function It is called the sinc function For this reason, e ill pursue it in more detail in this eample Specifically, e ill develop the Fourier a for t / transform of t () : otherise / it / i / i / i / i / it e e e a e e a X ( ) a e dt a a sin( / ) i i i / / a sin( / ) Writing this as X ( ) sin( / ) ( a ) ( a ) Sa( / ), e have the Fourier transform pair: / a for t / ( t) X ( ) ( a ) Sa( / ) otherise he Importance of the Rectangular Windo and Its Fourier ransform- the Sinc Function- o appreciate the importance of the rectangular indo and its Fourier transform, it helps to understand the folloing property of Fourier transform pairs: Property : Multiplication in one domain is equivalent to convolution in the other domain Eample 4 A random process { ( t); t (, )} can only be observed over a finite time [, ] Let for t [, ] t () denote a rectangular indo of length and height We can then rite the finite-time otherise

4 observation of { ( t); t (, )} as { ( t) ( t) ( t); t (, )} In ords, e have assumed that for t[, ] It follos from Property that X ( ) X W( ), here () t 4 equals zero it i i / i / i / it e e e e e i / W ( ) e dt Sa( / ) e i i i t No suppose that ( t) Asin( t ) here ~ Uniform[, ) hen it it X ( ) Asin( t ) e dt A Asin( t ) e dt i i Recalling the identity sin ( e e ) / i gives i( t) i( t) i e e e i t i( ) t X ( ) e i t dt e e i t dt e e i t dt i i i t Use the Fourier transform pair relation e ( ) gives i e X ( ) ( ) ( ) i his can be simplified by noting that i cos( / ) isin( / ) e i / Hence, It follos that X i( /) ( ) e ( ) ( ) X X W e W W i( /) ( ) ( )( ) ( ) ( ) Note: the factor is the appropriate scale factor in this convolution theorem In ords, the effect of the rectangular indo on the Fourier transform of the sinusoid is that the to delta functions have been replaced by to sinc functions Effectively, the spikes have smeared out For a small indo the smearing effect ill be large; hereas for a large indo it ill be more localized Remark 3 We have just alluded to hat, more generally is knon as the Heisenberg Uncertainty principle: he more you attempt to localize in one domain, the more uncertainty you have in the associated Fourier transform domain for [, ] Eample 5 he ideal lo pass filter (LPF) is the rectangular frequency indo W ( ) otherise Applying it gives X ( ) X ( ) W( ) Its effect in the time domain is:

5 5 ( t) (/ )( )( t) his time domain convolution ill smear out the details in o demonstrate this consider discrete-time hite noise ith sampling interval equal to second he analysis bandidth is then [, ) We ill apply an ideal LPF having a bandidth frequency /4 t () to this hite noise he results are shon at right As epected, the details of the hite noise have been reduced (ie smeared out) Figure 3 Plots of unfiltered and filtered hite noise he Unbiased Versus the Biased Lagged-Product Autocorrelation Estimator- he unbiased and biased lagged-product estimators of R () are: ( ub) RX ( ) ( t) ( t ) dt and ( b) RX ( ) ( t) ( t ) dt Using the linearity of E(*) it is easy to sho that: ( ub E R ) X ( ) RX( ) and ( b) E RX ( ) RX ( ) ( ) RX ( ) t for t [, ] he indo () t is a triangular indo of idth In fact, it can be obtained by otherise convolving the rectangular indo have for t [ /, / ] t () ith itself Hence, by the convolution theorem e otherise W ( ) Sa( / ) he effect of convolving this indo ith the PSD S ( ) is to smear out the detail in S( ) In other ords, it decreases the spectral resolution Remark 4 In addition to decreasing the spectral resolution, the rectangular indo also changes the nature of deterministic random processes hey are converted to regular random processes For eample, recall from Eample 4 that for the deterministic random process ( t) Asin( t ) ( t) Asin( t ) here i ~ Uniform[, ) e found that ( X( ) e /) ( ) ( ) Hence, it is natural to assume the

6 corresponding psd ill include these to delta functions We then found that i( /) X ( ) e W ( ) W ( ) And so, once again, it is natural to presume that the corresponding psd ill no longer include these delta function Rather, it ill be more ell-behaved, in the sense that this psd is actually a function of frequency; as opposed to the delta function that is not a regular function at all Recall that the psd is a [poer spectral density Its units are poer/hz he psd is analogous to a force density, hich e call pressure he units of pressure are force/unit-length One need to compute force by integrating pressure over a given length If that length includes a single point, this integral is zero We ill no address one of the most important theorems in relation to computation of the psd: My Version of the Wiener-Kinchin heorem Let t ( ) () t () t () be any regular random process ith psd S ( ) be a indoed version of, here the time indo as idth We shoed that its psd is: ( b) E RX ( ) RX ( ) ( ) RX ( ) For convenience, denote this triangular-indoed autocorrelation function as he corresponding psd is: ( ) ( ) i ( ) ( ) R ( ) () S R e d () ( ) ( ) On pp7-7 of the book it is shon that: S ( ) E X ( ) () his epression for the indoed psd is my version of the Wiener-Kinchin heorem he relation (s) gives an alternative to taking the Fourier transform of the lagged-product autocorrelation function Suppose that the indo length can be partitioned into n smaller indos, each having length Δ Denote the process associated ith the k th ( k indo as ) ( k () t Its Fourier transform is X ) ( ) We can then estimate () via the average: S (3) n ( ) ( k) ( ) X ( ) n k If e do not partition and use the entire indo, e have: ( ) S ( ) X( ) (4) he estimator (4) is called the periodogram In ords, you simply compute the Fourier transform of { ( t); t (, )}, and then take its magnitude-squared as the estimator of the psd No, an average of one is nothing to rite home about By partitioning and taking an average of n mod-squared Fourier transforms e can significantly reduce the uncertainty of the psd estimator Hoever, this comes at a price he sinc function associated ith a indo of size / nill be ider than the sinc function associated ith a indo of size by a factor of n In ords, e reduce the spectral resolution of the estimator It must be emphasized: he periodogram (4) is a horrible estimator of the psd! Let 6

7 he challenge is to determine the value of n such that the estimator ill have acceptably lo uncertainty, hile at the same time having reasonable spectral resolution Finally, it should be mentioned that this problem has been around for a long time, and that many methods of attacking it have been proposed One method is to use a specified overlap of the sub-indos For a 5% overlap, the second half of a given indo ould be the first half of the net indo Having more suindos allos one to use a larger sub-indo size Matlab offers this method (as ell as others) It is knon as Welch s method, and is the code pelchm 7 CONCLUSION: his lecture as packed ith elements of psd analysis here are entire graduate courses on this topic You should not be discouraged if your head is no spinning Re-read these lecture notes again and again Go to Google to help improve your understanding he psd is an indispensable topic in analysis of random processes For this reason, e ill return to it in the contet of our net topic, hich is linear systems