TABLE 14.2 Properties of the CT Fourier Transform If y f(r)= FGo), then: )= Superposition r[e()+5()=a(o)bE(o) (a)f(r) is even o)= f() o)-2f Ff(-)=Fe(jo) (a) Time Differentiation 7|z(=(o) Integration (=()+ Time shifting sf(t-a)=F(joJe"i f(jew=Fo -oo)I 7(s0o={1o-o)+ia+o) fmo=2ao)-|ao〗l Time convolution convolution frequency response. ropery, algorithms, since many systems can be specified directly by their impulse or basis for many signal-processil (frequency shifting)is useful for analyzing the performance of communication stems where different modulation formats are commonly used to shift spectral energy among different Fourier Spectrum of a CT Sampled Signal The operation of uniformly sampling a CT signal s(t)at every T seconds is characterized by Eq (14. 2), where 4()=∑()-n)=∑m)c-nr) (14.2) e 2000 by CRC Press LLC© 2000 by CRC Press LLC basis for many signal-processing algorithms, since many systems can be specified directly by their impulse or frequency response. Property 3 (frequency shifting) is useful for analyzing the performance of communication systems where different modulation formats are commonly used to shift spectral energy among different frequency bands. Fourier Spectrum of a CT Sampled Signal The operation of uniformly sampling a CT signal s(t) at every T seconds is characterized by Eq. (14.2), where d(t) is the CT impulse function defined earlier: (14.2) TABLE 14.2 Properties of the CT Fourier Transform Name If F f(t) = F(jw), then: Definition Superposition Simplification if: (a) f (t) is even (b) f (t) is odd Negative t Scaling: (a) Time (b) Magnitude Differentiation Integration Time shifting Modulation Time convolution Frequency convolution F j f t e dt f t F j e d t bf t aF j bF j j t j t w p w w w w w w ( ) = ( ) ( ) = ( ) [ ] ( ) + ( ) = ( ) + ( ) - -• • -• • Ú Ú 1 2 1 2 1 2 F af F j f t t dt F j j f t t dt w w w w ( ) = ( ) ( ) = ( ) • • Ú Ú 2 2 0 0 cos sin F f ( ) -t = *F ( ) jw F F f at a F j a af t aF j ( ) = Ê Ë Á ˆ ¯ ˜ ( ) = ( ) 1 w w F d dt f t j F j n n n ( ) È Î Í Í ˘ ˚ ˙ ˙ = ( w w ) ( ) F f x dx j F j F t ( ) È Î Í ˘ ˚ ˙ = ( ) + ( ) ( ) Ú-• 1 0 w w p d w F f t a F j e j a ( ) - = ( ) - w w F F F f t e F j f t t F j F j f t t j F j F j j t ( ) = - [ ] ( ) ( ) = - { } [ ] ( ) + + [ ( )] ( ) = - { } [ ] ( ) - + [ ( )] w w w w w w w w w w w w w 0 0 0 0 0 0 0 0 1 2 1 2 cos sin F - • • [ ] ( ) ( ) = ( ) ( ) - Ú 1 1 2 1 2 F jw w F j f t f t t d t – F f 1 (t)f ( )t F j F j d [ ] = ( ) ( ) - [ ] • • 2 Ú 1 2 1 2p l w l l – s t a as t t nT s nT t nT n a n ( ) = ( ) ( - ) = ( ) ( - ) = -• • = -• • Â Â d d