Lecture 5 Outline: Zero-Order Hold Sampling,

Upsampling and Downsampling

l Announcements:l Updated OHs:

● Me: MWF after class & by appt. (made by email or before/after class, late MWF ok) ● Wed 4PM-5PM (Malavika), 6PM-7PM (John); Thu 4PM-5PM (Georgia), Pack 3rd fl.

l HW 1 due Friday 4pm. HW 2 will be posted Fridayl Pauly’s FT handouts will be posted (different notation than Kahn’s)

l Review of Lecture 3 (Lecture 4 was Matlab tutorial)

l Sampling and Reconstruction with Zero-Order Hold

l Discrete-Time Upsampling and Reconstruction

l Digital Downsampling

Review of Lecture 3

l Digital to Analog Conversion

l Quantization

Discrete-ToContinuous

xd[n] xd(nTs)LPF

xr(t)DAC 0 2p-2p W

Xd(ejW)

w=W/Ts

0 2pTs

-2pTs

w

Xd(jw)Xr(jw)

-A

A

-A+D

-A+2D

-A+kD

……

x(t)

xQ(nTs)

Ts 2Ts0

x(t)ADC

Binary value of k

101101…

DAC101101… ][nxQd

)()][(][ niikAnxi

Qd -D+-= å

¥

-¥=

d

][nk

Continuous-TimeUnquantized

x(t) nTt =Anti-AliasingLowpass Filter

Bandlimits x(t) toprevent aliasing

Sampler

Quantizer

Discrete-TimeUnquantized

xd[n] = x(t)| t = nT

Discrete-TimeQuantized Representation

of xd[n] …

Digital Storage,Transmission,

Signal Processing, …

Discrete-TimeQuantized y[n] Reconstruction

SystemContinuous-Time

y(t)

Review Continued:Analog to Bits and Back

Continuous-TimeUnquantized

x(t) nTt =Anti-AliasingLowpass Filter

Bandlimits x(t) toprevent aliasing

Sampler

Quantizer

Discrete-TimeUnquantized

xd[n] = x(t)| t = nT

Discrete-TimeQuantized Representation

of xd[n] …

Digital Storage,Transmission,

Signal Processing, …

Discrete-TimeQuantized y[n] Reconstruction

SystemContinuous-Time

y(t)

Analog to Bits

Bits to Analog

0100100110010010000100001000100111…

0100100110010010000100001000100111…

ADC

DAC

Sampling with Zero-Order Hold

l Ideal sampling not possible in practicel delta function train cannot be realized

l In practice, ADC uses zero-order hold to produce xd[n] from x0(t)l Design uses switch, capacitor and resistors

l Reconstruction of x(t) from x0(t) removes h0(t) distortionl Multiplication in frequency domain by 1/H0(jw)

l Also use zero-order hold for reconstruction in practice

xs(t)ånd(t-nTs)

0 Ts 2Ts 3Ts 4Ts-3Ts -2Ts -Ts

x(t)´

xs(t) h0(t)1

Ts

x0(t) x0(t)xd[n] (Ts=1)

R1

R2

l Inserts L-1 zeros between each xd[n] value to get upsampledsignal xe[n]

l Compresses Xd(ejW) by L in W domain and repeats it every 2p/L; So Xe(ejW) is periodic every 2p/L

l Leads to less stringent reconstruction filter design than ideal LPF: zero-order hold often used

Discrete-Time Upsampling

0 1 2 3 4 0 L 2L 3L 4L

xd[n] xe[n]

0 p-pW

Xd(ejW)

WW-WW

WW=WTs

2p-2p W

Xe(ejW)

WW-WW

L L

… …

2pL

-2pL

xe[n]UpsampleBy L (­L)

xd[n]=x(nTs)

0w

X(jw)

W-W

Reconstruction of Upsampled Signal

l Pass through an ideal LPF Hi(ejW) to get xi[n] l xi[n]=ẋe[n]=x(nTs/L) if x(t) originally sampled at Nyquist rate (Ts<p/W)l Relaxes DAC filter requirements (approximate LPF/zero-order hold

reconstructs x(t) from xi[n]; better filter to reconstruct from xd[n] needed)

0p-p W

xd[n]=x(nTs)ÛXd(ejW)

WW-WW

Hi(ejW)

2p-2p W

Xe(ejW)

WW-WW

L L

0 L 2L 3L 4L

xe[n]

0 1 2 3 4

xd[n]

xi[n]xe[n]UpsampleBy L

xd[n]=x(nTs)

2pL

-2pL

0 L 2L 3L 4L

xi[n]

Xi(ejW) Û xi[n]xi[n]=ẋe[n]=x(nTs/L) if Ts<p/W

Xi(ejW)

ẋe[n]x(t)

DAC

Reconstruct x(t) from xi[n] by passing it through a DAC

… …

Digital Downsampling:Fourier Transform and Reconstruction

l Digital Downsamplingl Removes samples of x(nTs) for n≠MTsl Used under storage/comm. constraints

l Repeats Xd(ejW) every 2p/M and scales W axis by Ml This results in a periodic signal Xc(ejW) every 2pl Introduces aliasing if Xd(ejW) bandwidth exceeds p/Ml Can prefilter Xd(ejW) by LPF with bandwith p/M prior to

downsampling to avoid downsample aliasing

xc[n]DownsampleBy M

xd[n]=x(nTs)

0 1 2 3 40 123 …

0 p/M-p/M W’

Xd(ejW’)

0 p-p

Xc(ejW)

-2p 2p

……0 p/M-p/M W’

Xd(ejW’)

0 p-p

Xc(ejW)

-2p 2p

……

W=MW’W=MW’

Main Pointsl Sampling in practice uses a zero-order hold, whose distortion can be

completely removedl Zero order hold easily implemented with a switch, capacitor, and resistorsl Higher-order holds provide better performance in practice

l Can also use a zero-order hold in DAC for reconstructionl Allows for perfect recovery of original sampled signal with ideal filtersl In practice, use filters that approximate ideal filter but are easier to implement

l Upsampling of a discrete-time signal by L compresses its Fourier Transform by 1/Ll Equivalent to sampling L times faster if original sampling rate above Nyquist;

much easier to add zeros!!!!l Allows less stringent requirements on reconstruction filter than an ideal LPFl A zero-order hold is often used for the reconstruction

l Digital downsampling used when system constraints preclude storing/communicating samples every Ts seconds.