5.3 Effect of Sampling in the Frequency Domain gp(t)is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse at t=nT weighted by the sampled value ga(nT)of ga(t)at that instant t=nT &a(0 8p) 8a() 0 2T-T0T2T、 8a(4D
§5.3 Effect of Sampling in the Frequency Domain • gp(t) is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse at t = nT weighted by the sampled value ga(nT) of ga(t) at that instant t=nT
5.3 Effect of Sampling in the Frequency Domain The impulse train gp(t)can be expressed as 0-豆n From the frequency-shifting property of the CTFT,the CTFT of eirkiga(t)is given by Ga(j(-kT))
§5.3 Effect of Sampling in the Frequency Domain • The impulse train gp(t) can be expressed as ( ) ( ) 1 g t e g t a k j kt T p T ⋅ = ∑ ∞ =−∞ Ω • From the frequency-shifting property of the CTFT, the CTFT of ejΩTktga(t) is given by Ga(j(Ω - kΩT))
5.3 Effect of Sampling in the Frequency Domain Hence,the CTFT of gp(t)is given by Gm(2)=,∑GaU2-k2r》 Therefore,Go(js)is a periodic function of consisting of a sum of shifted and scaled replicas of Ga(j),shifted by integer multiples of r and scaled by 1/T
§5.3 Effect of Sampling in the Frequency Domain • Hence, the CTFT of gp(t) is given by ∑ ( ) ∞ =−∞ Ω = Ω − Ω k a T T p G ( j ) G j( k ) 1 • Therefore, Gp(jΩ) is a periodic function of Ω consisting of a sum of shifted and scaled replicas of Ga(jΩ) , shifted by integer multiples of ΩT and scaled by 1/T
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 G(j),and each of the remaining terms are the frequency translated portions of Gp(j) The frequency range s2≤ 2 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(jΩ) , and each of the remaining terms are the frequency translated portions of Gp(jΩ) • The frequency range 2 2 T ΩT ≤ Ω − Ω ≤ is called the baseband or Nyquist band
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 G.(j) 0 The spectrum P(j)of p(t)having a sampling period T=2m/r is indicated below P(jQ) 1 0 r 227 391
§5.3 Effect of Sampling in the Frequency Domain • Assume ga(t) is a band-limited signal with a CTFT Ga(jΩ) as shown below • The spectrum P(jΩ) of p(t) having a sampling period T=2π/ΩT is indicated below