1. Discrete-Time Upsamplingw
X(jw) W -W
xd[n]=x(nTs)
xe[n]
Upsample By L (L)
Inserts L-1 zeros between each xd[n] value to get upsampled signal xe[n]
Compresses Xd(ejW) by L in W domain and repeats it every 2p/L; So Xe(ejW) is periodic every 2p/L
Leads to less stringent reconstruction filter design than ideal LPF: zero-order hold often used 2p
-2p W
Xe(ejW) WW
-WW p -p W
Xd(ejW) WW
-WW
WW=WTs
xd[n]
xe[n] … … 1 2 3 4
-2p L L 2L 3L 4L 2p L L L

2. Reconstruction of Upsampled Signal
xd[n]=x(nTs)
xe[n]
Upsample
By L
x(t)
DAC
Reconstruct x(t) from xi[n] by passing it through a DAC
xi[n]
Hi(ejW)
ẋe[n]
Pass through an ideal LPF Hi(ejW) to get xi[n]
xi[n]=ẋe[n]=x(nTs/L) if x(t) originally sampled at Nyquist rate (Ts<p/W)
Relaxes DAC filter requirements (approximate LPF/zero-order hold reconstructs x(t) from xi[n]; better filter to reconstruct from xd[n] needed) 2p
-2p W
Xe(ejW) WW
-WW L 2L 3L 4L
xi[n]
Xi(ejW)  xi[n]
xi[n]=ẋe[n]=x(nTs/L) if Ts<p/W
Xi(ejW)
xd[n]=x(nTs)Xd(ejW) 2p L
-2p … … W p -p
-WW WW L L
xd[n]
xe[n] 1 2 3 4 L 2L 3L 4L