1. Lecture 6 Outline: Upsampling and Downsampling
Announcements:
HW 1 due today 4pm. HW 2 will be posted later today
Review of Last Lecture
Discrete-Time Upsampling
Reconstruction of Upsampled Signal
Discrete-Time Downsampling
Reconstruction of Downsampled Signal

2. Review of Last Lecture Sampling with Zero-Order Hold
nd(t-nTs)
xs(t)
xd[n] (Ts=1) Ts
2Ts
3Ts
4Ts
-3Ts
-2Ts
-Ts
x(t)
xs(t)
h0(t)
x0(t)
x0(t)  1 Ts R1 R2
Ideal sampling not possible in practice
In practice, ADC uses zero-order hold to produce xd[n] from x0(t)
Reconstruction of x(t) from x0(t) removes h0(t) distortion
Multiplication in frequency domain by TP 𝝎 𝝎 𝒔 /H0(jw)
Also use zero-order hold for reconstruction in practice
xd[n]=x(nTs)
h0(t)
Discrete To
Continuous
xd(t)
x0(t)
Hr(jw)
xr(t)=x(t) 1 Ts

3. Zero-order hold model  nd(t-nTs) x(t) xs(t) h0(t) x0(t) Hr(jw)
xr(t)=x(t)  1 T
Ho(jw)Hr(jw)=T∙P 𝝎 𝝎 𝒔
Zero-order hold model
X(jw) T
|Hr(jw)|
Xs(jw)
|H0(jw)|
|X0(jw)|

4. 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

5. Reconstruction of Upsampled Signal
More stringent LPF (DT)
Less stringent LPF
xd[n]=x(nTs)
xe[n]
xi[n]
Upsample
By L
x(t)
DAC
Reconstruct x(t) from xi[n] by passing it through a DAC
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(ejW)
xd[n]=x(nTs)Xd(ejW) 2p L
-2p … …
xi[n]=ẋe[n]=x(nTs/L) if Ts<p/W W p -p
-WW WW L L
xd[n]
xe[n] 1 2 3 4 L 2L 3L 4L

6. Discrete-Time Downsampling: Fourier Transform and Reconstruction
Digital Downsampling
Removes samples of x(nTs) for n≠MTs
Used under storage/comm. constraints
Repeats Xd(ejW) every 2p/M and scales W axis by M
This results in a periodic signal Xc(ejW) every 2p
Introduces aliasing if Xd(ejW) bandwidth exceeds p/M
Can prefilter Xd(ejW) by LPF with bandwith p/M prior to downsampling to avoid downsample aliasing 1 2 3 …
xd[n]=x(nTs)
xc[n]
Downsample
By M 1 2 3 4
p/M
-p/M W’
Xd(ejW’) p -p
Xc(ejW)
-2p 2p …
p/M
-p/M W’
Xd(ejW’) p -p
Xc(ejW)
-2p 2p …
W=MW’
W=MW’

7. Main Points
Upsampling of a discrete-time signal by L compresses its Fourier Transform by 1/L
Equivalent to sampling L times faster if original sampling rate above Nyquist; much easier to add zeros!!!!
Allows less stringent requirements on reconstruction filter than an ideal LPF
Reconstruct upsampled signal by passing it through an ideal/approximate LPF
Upsampling eases the requirements on the reconstruction filter; a zero-order hold often used instead of a LPF
Reconstructed signal is sinc interpolation of signal sampled every Ts and upsampled at rate L or signal sampled every Ts/L
Digital downsampling used when system constraints preclude storing/communicating samples every Ts seconds.
Removes samples of x(nTs) for n≠MTs; Repeats Xd(ejW) every 2p/M
Introduces aliasing if Xd(ejW) bandwidth exceeds p/M