85.3 Effect of Sampling in the Frequency Domain The impulse train gp(t) therefore can be expressed as gn(t) SejS2Tkt 8a() k=-0 From the frequency-shifting property of the ctft, the ctft of ejsrktga (t is given e by Ga((Q2-KQ2D)
§5.3 Effect of Sampling in the Frequency Domain • The impulse train gp (t) therefore can be expressed as ( ) ( ) 1 g t e g t a k j kt T p T = =− • From the frequency-shifting property of the CTFT, the CTFT of ejTktga (t) is given by Ga (j( - kT ))
85.3 Effect of Sampling in the Frequency Domain Hence an alternative form of the ctft of g (t)is given by Gn(7G2((9-k27) Therefore GngjQ2) is a periodic function of Q consisting of a sum of shifted and scaled replicas of ga(j 2), shifted by integer multiples of S and scaled by 1/T
§5.3 Effect of Sampling in the Frequency Domain • Hence, an alternative form of the CTFT of gp (t) is given by ( ) =− = − k a T T p G ( j ) G j( k ) 1 • Therefore, Gp (j) is a periodic function of consisting of a sum of shifted and scaled replicas of Ga (j) , shifted by integer multiples of T and scaled by 1/T
85.3 Effect of Sampling in the Frequency Domain The term on the rhs of the previous equation for k=0 is the baseband portion of gpgjS2), and each of the remaining terms are the frequency translated portions of gjQ2) The frequency range 2 is called the baseband or Nyquist band
§5.3 Effect of Sampling in the Frequency Domain • The term on the RHS of the previous equation for k = 0 is the baseband portion of Gp (j) , and each of the remaining terms are the frequency translated portions of Gp (j) • The frequency range 2 2 T T − is called the baseband or Nyquist band
85.3 Effect of Sampling in the Frequency Domain Assume ga(t) is a band-limited signal with a CtFT GagjQ2)as shown below G,(Q) The spectrum P(jQ2)of p(t) having a sampling a period T=2/T is indicated below PQ) T 29
§5.3 Effect of Sampling in the Frequency Domain • Assume ga (t) is a band-limited signal with a CTFT Ga (j) as shown below • The spectrum P(j) of p(t) having a sampling period T=2/T is indicated below
85.3 Effect of Sampling in the Frequency Domain Two possible spectra of gp(@2)are shown below ) T 1/T 29 39 g T
§5.3 Effect of Sampling in the Frequency Domain • Two possible spectra of Gp (j) are shown below