85.3 Effect of Sampling in the Frequency domain The impulse train gn(t) therefore can be expressed as ∑eAh ga(t) From the frequency-shifting property of the ctft, the ctft of eisrktga(t)is given byG1(2-k4)
§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 ⋅ = ∑ ∞ =−∞ Ω • From the frequency-shifting property of the CTFT, the CTFT of ejΩTktga (t) is given by Ga (j(Ω - kΩT ))
85.3 Effect of Sampling in the Frequency domain Hence, an alternative form of the ctft of gp(t) is given by Gn()=∑G(9-n) 〓-0 Therefore, Gngje2)is a periodic function of a consisting of a sum of shifted and scaled replicas of Q2), shifted by integer multiples of OT 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 ∑ ( ) ∞ =−∞ 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
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 ofgngj 2), and each of the remaining terms are the frequency translated portions ofGngjQ2) · The frequency range Q< 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
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(ja) 90Ω The spectrum pgo)of p(t) having a sampling period T=2T/T is indicated below PS) Q 29
§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
85.3 Effect of Sampling in the Frequency domain Two possible spectra ofGngj Q2) are shown below G,(S) T R Q 1/T 39
§5.3 Effect of Sampling in the Frequency Domain • Two possible spectra of Gp (jΩ) are shown below