fourier transform properties

 

The Fourier transform of a real, continuous-time signal x(t) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWG4bGaaiikaiaadshacaGGPaaaaa@398F@  is a complex-valued function defined by

X(ω)   =    ∫ −∞ ∞ x(t)      e −jω t    dt MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYhH8qipfea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiaacIcacqaHjpWDcaGGPaGaaGjbVlabg2da9iaaysW7daWdXbqaaiaadIhacaGGOaGaamiDaiaacMcacaaMe8oaleaacqGHsislcqGHEisPaeaacqGHEisPa0Gaey4kIipakiaaykW7caWGLbWaaWbaaSqabeaacqGHsislcaWGQbGaeqyYdCNaaGPaVlaadshaaaGccaaMe8Uaamizaiaadshaaaa@54F6@

 

where w is a real variable (frequency, in radians/second) and j= −1 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYhH8qipfea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabg2da9maakaaabaGaeyOeI0IaaGymaaWcbeaaaaa@3925@ .

 

A plot of |X(ω)| MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYhH8qipfea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIfacaGGOaGaeqyYdCNaaiykaiaacYhaaaa@3B70@  vs w is called the magnitude spectrum of x(t) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWG4bGaaiikaiaadshacaGGPaaaaa@398F@ , and a plot of ∢ X(ω) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYhH8qipfea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSOiImLaaGPaVlaadIfacaGGOaGaeqyYdCNaaiykaaaa@3C23@  vs w is called the phase spectrum of x(t) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWG4bGaaiikaiaadshacaGGPaaaaa@398F@ . These plots, particularly the magnitude spectrum, provide a picture of the frequency composition of the signal. For a real signal, the magnitude spectrum is an even function of frequency, |X(−ω)|   =   |X(ω)| MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaGG8bGaamiwaiaacIcacqGHsislcqaHjpWDcaGGPaGaaiiFaiaaysW7cqGH9aqpcaaMe8UaaiiFaiaadIfacaGGOaGaeqyYdCNaaiykaiaacYhaaaa@4753@ . The phase spectrum will be plotted for angles in the principle range −π≤∢X(ω)≤π MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqGHsislcqaHapaCcqGHKjYOcqWIIiYucaWGybGaaiikaiabeM8a3jaacMcacqGHKjYOcqaHapaCaaa@433C@ , and the choice between −π MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqGHsislcqaHapaCaaa@38EA@   and π MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqaHapaCaaa@37FD@   will be made so that ∢X(−ω)=−∢X(ω) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqWIIiYucaWGybGaaiikaiabgkHiTiabeM8a3jaacMcacqGH9aqpcqGHsislcqWIIiYucaWGybGaaiikaiabeM8a3jaacMcaaaa@4376@ , for ω≠0 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqaHjpWDcqGHGjsUcaaIWaaaaa@3A8E@ . That is, the phase spectrum will be shown as an odd function of frequency, except that ∢X(0) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqWIIiYucaWGybGaaiikaiaaicdacaGGPaaaaa@3A58@  might not be zero.

 

For a number of signals of interest, the Fourier transform integral does not converge in the usual sense of elementary calculus. Some of these signals can be treated in a consistent fashion by admitting Fourier transforms that contain so-called generalized functions. For example, if x(t)=u(t) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWG4bGaaiikaiaadshacaGGPaGaeyypa0JaamyDaiaacIcacaWG0bGaaiykaaaa@3DE1@ , the unit-step signal, then

X(ω)=    1 jω    +   π δ(ω) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWGybGaaiikaiabeM8a3jaacMcacqGH9aqpcaaMe8+aaSaaaeaacaaIXaaabaGaamOAaiabeM8a3baacaaMe8Uaey4kaSIaaGjbVlabec8aWjaaykW7cqaH0oazcaGGOaGaeqyYdCNaaiykaaaa@4C6B@

 

where δ(ω) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqaH0oazcaGGOaGaeqyYdCNaaiykaaaa@3B0B@  is the unit impulse. For such a Fourier transform, we treat impulse components as separate in computing the magnitude spectrum since an impulse is zero at all values of ω MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqaHjpWDaaa@380D@  but one, though admittedly something very special happens at that one point. Thus for the unit-step example,

|X(ω)|   =    1 |ω|    +   π δ(ω) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaGG8bGaamiwaiaacIcacqaHjpWDcaGGPaGaaiiFaiaaysW7cqGH9aqpcaaMe8+aaSaaaeaacaaIXaaabaGaaiiFaiabeM8a3jaacYhaaaGaaGjbVlabgUcaRiaaysW7cqaHapaCcaaMc8UaeqiTdqMaaiikaiabeM8a3jaacMcaaaa@5109@

 

In plotting the magnitude spectrum, we indicate the impulse term using an arrow. For the phase spectrum display, we ignore any impulse term, which contributes angle, the angle of the “area” of the impulse, at only one value of ω MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqaHjpWDaaa@380D@ . Thus for the unit-step signal the phase spectrum is given by

∢X(ω)   =   { −π/2   , ω>0    π/2   , ω<0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqWIIiYucaWGybGaaiikaiabeM8a3jaacMcacaaMe8Uaeyypa0JaaGjbVpaaceaabaqbaeqabiqaaaqaaiabgkHiTiabec8aWjaac+cacaaIYaGaaGjbVlaacYcacaaMf8UaeqyYdCNaeyOpa4JaaGimaaqaaiaaysW7cqaHapaCcaGGVaGaaGOmaiaaysW7caGGSaGaaGzbVlabeM8a3jabgYda8iaaicdaaaaacaGL7baaaaa@5833@

 

 

The applet below illustrates properties of the magnitude and phase spectra of signals, and the effect on the spectra of typical operations on signals. Select a signal from the provided signals, and the corresponding magnitude and phase spectra will be displayed, both in mathematical terms and graphically. Then select an operation and the resulting signal and its spectra are displayed.

 

Note that impulses are shown as arrows, but the area is not indicated. Also, amplitude scaling an impulse should be interpreted as area scaling. Finally, the cosine pulse is chosen so that the pulse begins and ends at a zero crossing of the cosine.

 

 

 

Signal

x(t)

X(w)

unit impulse

δ(t) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqaH0oazcaGGOaGaamiDaiaacMcaaaa@3A37@

1 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaaIXaaaaa@36FB@

unit step

u(t) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWG1bGaaiikaiaadshacaGGPaaaaa@398C@

1 jω    +   π δ(ω) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaadaWcaaqaaiaaigdaaeaacaWGQbGaeqyYdChaaiaaysW7cqGHRaWkcaaMe8UaeqiWdaNaaGPaVlabes7aKjaacIcacqaHjpWDcaGGPaaaaa@45D6@

exponential

e −at u(t)   , a>0 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiDaaaakiaadwhacaGGOaGaamiDaiaacMcacaaMe8UaaiilaiaaywW7caWGHbGaeyOpa4JaaGimaaaa@43EC@

1 a+jω MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8VfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaGymaaqaaiaadggacqGHRaWkcaWGQbGaeqyYdChaaaaa@3A6B@

 t-exponential

t  e −at u(t)   , a>0 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWG0bGaaGPaVlaadwgadaahaaWcbeqaaiabgkHiTiaadggacaWG0baaaOGaamyDaiaacIcacaWG0bGaaiykaiaaysW7caGGSaGaaGzbVlaadggacqGH+aGpcaaIWaaaaa@4670@

1 (a+jω) 2 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8VfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaGymaaqaaiaacIcacaWGHbGaey4kaSIaamOAaiabeM8a3jaacMcadaahaaWcbeqaaiaaikdaaaaaaaaa@3CAD@

tent

e −a|t|    , a>0 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaaiiFaiaadshacaGG8baaaOGaaGjbVlaacYcacaaMf8Uaamyyaiabg6da+iaaicdaaaa@42A0@

2a a 2 + ω 2 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8VfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaGOmaiaadggaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaeqyYdC3aaWbaaSqabeaacaaIYaaaaaaaaaa@3C3F@

Gaussian

e − a 2 t 2 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWGLbWaaWbaaSqabeaacqGHsislcaWGHbWaaWbaaWqabeaacaaIYaaaaSGaamiDamaaCaaameqabaGaaGOmaaaaaaaaaa@3C02@

π a   e − ω 2 /4 a 2 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaadaWcbaWcbaWaaOaaaeaacqaHapaCaWqabaaaleaacaWGHbaaaOGaaGPaVlaadwgadaahaaWcbeqaaiabgkHiTiabeM8a3naaCaaameqabaGaaGOmaaaaliaac+cacaaI0aGaamyyamaaCaaameqabaGaaGOmaaaaaaaaaa@42C2@

 

 

For each signal, you can select an operation and the effects of the operation on the signal and its spectra are displayed, both graphically and in terms of analytical expressions. The available operations are described in the table below.

 

 

Operation on x(t)

Resulting signal y(t)

Y(w)

amplitude scale

b x(t) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWGIbGaaGPaVlaadIhacaGGOaGaamiDaiaacMcaaaa@3C01@

b X(ω) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWGIbGaaGPaVlaadIfacaGGOaGaeqyYdCNaaiykaaaa@3CB5@

time shift

x(t−T) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWG4bGaaiikaiaadshacqGHsislcaWGubGaaiykaaaa@3B55@

e −jω T    X(ω) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWGLbWaaWbaaSqabeaacqGHsislcaWGQbGaeqyYdCNaaGPaVlaadsfaaaGccaaMe8UaamiwaiaacIcacqaHjpWDcaGGPaaaaa@42FE@

time scale

x(at)   , a≠0 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8VfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamyyaiaadshacaGGPaGaaGjbVlaacYcacaaMf8UaamyyaiabgcMi5kaaicdaaaa@4083@

1 |a|  X(ω/a) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8VfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaGymaaqaaiaacYhacaWGHbGaaiiFaaaacaaMc8UaamiwaiaacIcacqaHjpWDcaGGVaGaamyyaiaacMcaaaa@3FF4@

time reverse

x(− t) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWG4bGaaiikaiabgkHiTiaaykW7caWG0bGaaiykaaaa@3C07@

X(−ω) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8VfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaGGOaGaeyOeI0IaeqyYdCNaaiykaaaa@3A0C@

derivative

dx(t) dt MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaadaWcaaqaaiaadsgacaWG4bGaaiikaiaadshacaGGPaaabaGaamizaiaadshaaaaaaa@3C6A@

jω X(ω) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWGQbGaeqyYdCNaaGPaVlaadIfacaGGOaGaeqyYdCNaaiykaaaa@3E8A@

running integral

∫ −∞ t x(σ)   dσ MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaadaWdXbqaaiaadIhacaGGOaGaeq4WdmNaaiykaiaaysW7caWGKbGaeq4WdmhaleaacqGHsislcqGHEisPaeaacaWG0baaniabgUIiYdaaaa@4452@

1 jω    X(ω)   +   πX(0)δ(ω) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8VfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaGymaaqaaiaadQgacqaHjpWDaaGaaGjbVlaadIfacaGGOaGaeqyYdCNaaiykaiaaysW7cqGHRaWkcaaMe8UaeqiWdaNaamiwaiaacIcacaaIWaGaaiykaiabes7aKjaacIcacqaHjpWDcaGGPaaaaa@4BA7@

 

 

In justifying the spectra resulting from an operation, you might need to apply standard properties of impulses, in particular,

δ(−ω)=δ(ω)   ; δ(aω)=  1 a δ(ω),   a>0    MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqaH0oazcaGGOaGaeyOeI0IaeqyYdCNaaiykaiabg2da9iabes7aKjaacIcacqaHjpWDcaGGPaGaaGjbVlaacUdacaaMf8UaeqiTdqMaaiikaiaadggacqaHjpWDcaGGPaGaeyypa0JaaGPaVpaalaaabaGaaGymaaqaaiaadggaaaGaeqiTdqMaaiikaiabeM8a3jaacMcacaGGSaGaaGjbVlaadggacqGH+aGpcaaIWaGaaGjbVdaa@5AD2@

 

and, if X(ω) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWGybGaaiikaiabeM8a3jaacMcaaaa@3A43@  is an ordinary function that is continuous at ω=0 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqaHjpWDcqGH9aqpcaaIWaaaaa@39CD@ ,

X(ω) δ(ω)   =   X(0) δ(ω) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWGybGaaiikaiabeM8a3jaacMcacaaMc8UaeqiTdqMaaiikaiabeM8a3jaacMcacaaMe8Uaeyypa0JaaGjbVlaadIfacaGGOaGaaGimaiaacMcacaaMc8UaeqiTdqMaaiikaiabeM8a3jaacMcaaaa@4DFE@

 

For example, the time shifted unit-step signal, u(t−T) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaWG1bGaaiikaiaadshacqGHsislcaWGubGaaiykaaaa@3B52@ , has the Fourier transform

e −jωT [    1 jω    +   π δ(ω)]   =    e −jωT jω    +   π  e −jωT δ(ω)    =    e −jωT jω    +   π δ(ω) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@85AE@

 

Furthermore, derivatives of discontinuous signals must be interpreted in the generalized sense. For example, the derivative of the unit step is the unit impulse, and the corresponding transform operation gives

jω   [    1 jω    +   π δ(ω)]   =   1   +   jπ ω δ(ω)    =   1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakqaaeeqaaiaadQgacqaHjpWDcaaMe8Uaai4waiaaysW7daWcaaqaaiaaigdaaeaacaWGQbGaeqyYdChaaiaaysW7cqGHRaWkcaaMe8UaeqiWdaNaaGPaVlabes7aKjaacIcacqaHjpWDcaGGPaGaaiyxaiaaysW7cqGH9aqpcaaMe8UaaGymaiaaysW7cqGHRaWkcaaMe8UaamOAaiabec8aWjaaykW7cqaHjpWDcaaMc8UaeqiTdqMaaiikaiabeM8a3jaacMcacaaMe8oabaGaeyypa0JaaGjbVlaaigdaaaaa@6781@

 

(An impulse of zero area is interpreted as no impulse.)

 

The derivative of an impulse is a “doublet.” This generalized signal is shown as an up/down arrow, but the mathematical properties of doublets are beyond our scope. Also, the running-integral operation on a signal typically yields a signal that has generalized functions in its Fourier transform. As an important example, the running integral of a unit step is a unit ramp, a signal whose transform involves a doublet.

Pick a signal type: Pick a signal operation:

 

Exercises

 

 

 

d dt   e −at u(t)   =   −a  e −at u(t)   +   δ(t) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaadaWcaaqaaiaadsgaaeaacaWGKbGaamiDaaaacaaMc8UaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadshaaaGccaWG1bGaaiikaiaadshacaGGPaGaaGjbVlabg2da9iaaysW7cqGHsislcaWGHbGaaGPaVlaadwgadaahaaWcbeqaaiabgkHiTiaadggacaWG0baaaOGaamyDaiaacIcacaWG0bGaaiykaiaaysW7cqGHRaWkcaaMe8UaeqiTdqMaaiikaiaadshacaGGPaaaaa@5888@

 

 

·        How do you distinguish between the time reversal and the amplitude reversal (amplitude scale by −1 MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqGHsislcaaIXaaaaa@37E8@  ) of a signal based on the resulting spectra? (For example, beginning with     e −at u(t) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaaMe8UaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadshaaaGccaWG1bGaaiikaiaadshacaGGPaaaaa@3F06@  consider the spectra of     e at u(−t) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacaaMe8UaamyzamaaCaaaleqabaGaamyyaiaadshaaaGccaWG1bGaaiikaiabgkHiTiaadshacaGGPaaaaa@3F06@  and − e −at u(t) MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVeYdOipfYlH8qipiY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeaabaWaaeaaeaaakeaacqGHsislcaWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiDaaaakiaadwhacaGGOaGaamiDaiaacMcaaaa@3E66@ .)