Discrete-Time Upsampling w 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

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