1. EE 102b: Signal Processing and Linear Systems II
Midterm Review
Signals and
Systems

2. Sampling and Reconstruction vs
Sampling and Reconstruction vs. Analog-to-Digital and Digital-to-Analog Conversion
Sampling: converts a continuous-time signal to a continuous-time sampled signal
Reconstruction: converts a sampled signal to a continuous-time signal.
Analog-to-digital conversion: converts a continuous-time signal to a discrete-time quantized or unquantized signal
Digital-to-analog conversion. Converts a discrete-time quantized or unquantized signal to a continuous-time signal. Ts
2Ts
3Ts
4Ts
-3Ts
-2Ts
-Ts Ts
2Ts
3Ts
4Ts
-3Ts
-2Ts
-Ts 1 2 3 4 -3 -2 -1
Each level can
be represented
by 0s and 1s 1 2 3 4 -3 -2 -1

3. Sampling Sampling (Time): =
Sampling (Frequency): x(t)p(t) X(jw)*P(jw)/(2p)
Analog-to-Digital Conversation (ADC)
Setting xd[n]=x(nTs) yields Xs(ejW) with W=wTs
x(t) =
p(t)=nd(t-nTs)
xs(t) Ts
2Ts
3Ts
4Ts
-3Ts
-2Ts
-Ts
-3Ts
-2Ts
-Ts Ts
2Ts
3Ts
4Ts
𝟐𝝅 𝑻𝒔
X(jw) * =
nd(w-(2pn/Ts))
Xs(jw) 1
1/Ts
-2p Ts 2p Ts
-2p Ts 2p Ts

4. Reconstruction Frequency Domain: low-pass filter (Ts=p/W)
Time Domain: sinc interpolation
Digital-to-Analog Conversation (DAC)
LPF applied to Xs(ejW) and then converted to continuous time (w=W/Ts) recovers sampled signal
H(jw)
Xs(jw)
-2p Ts
Xs(jw) 2p
Xr(jw)
H(jw) Ts -W W w w
-2p Ts 2p Ts

5. Nyquist Sampling Theorem
A bandlimited signal [-W,W] radians is completely described by samples every Tsp/W secs.
The minimum sampling rate for perfect reconstruction, called the Nyquist rate, is W/p samples/second
If a bandlimited signal is sampled below its Nyquist rate, distortion (aliasing occurs)
X(jw)
X(jw)
Xs(jw) -W W
-2W W W
2W=2p/Ts

6. Quantization x(t) xQ(nTs) A … -A+kD -A+2D -A+D -ATs
2Ts …
Divide amplitude range [-A,A] into 2N levels, {-A+kD}, k=0,…2N-1
Map x(t) amplitude at each Ts to closest level, yields xQ(nTs)=xQ[n]
Convert k to its binary representation (N bits); converts xQ[n] to bits

7. Analog to Bits and Back Analog to Bits Bits to Analog ADC DAC …
Bits to Analog
DAC …

8. 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 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

9. 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)|

10. 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

11. Reconstruction of Upsampled Signal
More stringent LPF than for Xd(ejW)
Less stringent analog LPF
than to reconstruct from xd[n]
xd[n]=x(nTs)
xe[n]
Upsample
By L
x(t)
DAC
Reconstruct x(t) from xi[n] by passing it through a DAC
=ẋe[n]
xi[n]=ẋe[n]=x(nTs/L) if Ts<p/W
xi[n]
Hi(ejW)
xr(t)≠x(t)
xd[n]
xi[n]
Discrete
To Cts
Ha(jw)
x(t)
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)
Ha(ejW)
xd[n]=x(nTs)Xd(ejW)
Ha(ejW) 2p L
-2p … … W p -p
-WW WW L L
xd[n]
xe[n] 1 2 3 4 L 2L 3L 4L

12. Proof that xi[n]=ẋe[n]
Xi(ejW)
=Xe(ejW) 2p
-2p W
Xd(ejW)
W=wTs Ts w
Xs(jw)
X(jw) -W
Ts/L 2p
-2p W
Xe(ejW)
W=wTs/L w
X(jw) -W
2pL Ts
-2pL
Xs(jw) .
Xi(ejW)=Lrect[WL/(2p)] 2p
-2p
2p/L
-2p/L W
Xe(ejW) …
=Xe(ejW) L 2L 3L 4L
xe[n] 1 2 3 4
xd[n]
𝑥 𝑒 𝑛 ≜𝑥 𝑛 𝑇 𝑠 𝐿 𝑋 𝑠 𝑇𝑠 𝐿 𝑗𝜔 𝑋 𝑒 𝑒 𝑗Ω = 𝑋 𝑒 𝑒 𝑗𝜔𝑇𝑠/𝐿 = 𝑋 𝑒 𝑒 𝑗𝜔𝑇𝑠/𝐿 𝐿rect 𝜔𝑇𝑠𝐿 2𝜋
𝑋 𝑒 𝑒 𝑗𝜔𝑇𝑠/𝐿 𝐿rect 𝜔𝑇𝑠𝐿 2𝜋 = 𝑋 𝑑 𝑒 𝑗𝜔𝑇𝑠/𝐿 𝐿rect 𝜔𝑇𝑠𝐿 2𝜋
=𝑋𝑒 𝑒 𝑗𝜔𝑇𝑠 𝐿rect 𝜔𝑇𝑠𝐿 2𝜋 =𝑋𝑒 𝑒 𝑗Ω 𝐿rect Ω𝐿 2𝜋 =𝑋𝑖 𝑒 𝑗Ω
xi[n]=ẋe[n]=x(nTs/L)
if Ts<p/W

13. Digital Downsampling: Fourier Transform and Reconstruction
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’

14. W=wTs W=wMTs Xs(jw) Xd(ejW) w W Xs(jw) Xc(ejW) W2p
-2p W
Xd(ejW)
W=wTs Ts w
Xs(jw)
X(jw) -W
𝑋𝑑 𝑒 𝑗Ω = 1 𝑇𝑠 𝑘=−∞ ∞ 𝑋 𝑗 Ω−2𝜋𝑘 𝑇𝑠
MTs 2p
-2p W
Xc(ejW)
W=wMTs w
X(jw) -W
Xs(jw)
𝑋𝐶 𝑒 𝑗Ω = 1 𝑀𝑇𝑠 𝑙=−∞ ∞ 𝑋 𝑗 Ω−2𝜋𝑙 𝑀𝑇𝑠
𝑙=𝑘𝑀+𝑚: 𝑙=−∞ ∞ = 𝑚=0 𝑀−1 𝑘=−∞ ∞
𝑋𝐶 𝑒 𝑗Ω = 1 𝑀𝑇𝑠 𝑙=−∞ ∞ 𝑋 𝑗 Ω−2𝜋𝑙 𝑀𝑇𝑠 = 1 𝑀 𝑚=0 𝑀−1 1 𝑇𝑠 𝑘=−∞ ∞ 𝑋 𝑗 Ω−2𝜋(𝑘𝑀+𝑚) 𝑀𝑇𝑠 = 1 𝑀 𝑚=0 𝑀−1 𝑋𝑑 𝑒 𝑗(Ω−2𝜋𝑚)/𝑀
(9)
𝑋𝑐 𝑒 𝑗Ω = 1 𝑀 𝑚=0 𝑀−1 𝑋𝑑 𝑒 𝑗(Ω−2𝜋𝑚)/𝑀
Repeats Xd(ejW) every 2p/M and scales W axis by M 1 2 3 4
xc[n]=xd[nM] 2p
-2p p -p W
Xc(ejW)
xd[n] 1 2 3 … 2p
-2p
-p/M
p/M W
Xd(ejW)

15. Communication System Block Diagram
Modulator (Transmitter)
Demodulator
(Receiver)
Channel
Modulator (Transmitter) converts message signal or bits into format appropriate for channel transmission (analog signal).
Channel introduces distortion, noise, and interference.
Demodulator (Receiver) decodes received signal back to message signal or bits.
Focus on modulators with s(t) at a carrier frequency wc. Allows allocation of orthogonal frequency channels to different users

16. Amplitude Modulation DSBSC and SSB
Double sideband suppressed carrier (DSBSC)
Modulated signal is s(t)=m(t)cos(wct)
Signal bandwidth (bandwidth occupied in positive frequencies) is 2W
Redundant information: can either transmit upper sidebands (USB) only or lower sidebands (LSB) only and recover m(t)
Single sideband modulation (SSB); uses 50% less bandwidth (less $$$)
Demodulator for DSBSC/SSB: multiply by cos(wct) and LPF W 2W
USB
LSB w -W W
-wc w wc
2wc
-2wc X
cos(wct)
s(t) wc
-wc

17. AM Radio + + cos(wct) s(t)=[A+kam(t)]coswct ka A m(t)1 W -W wc
-wc
m(t) + + X
Broadcast AM has s(t)=[1+kam(t)]cos(wct) with [1+kam(t)]>0
Constant carrier cos(wct) carriers no information; wasteful of power
Can recover m(t) with envelope detector (diode, resistors, capacitor)
Modulated signal has twice bandwidth W of m(t), same as DSBSC
1/(2pwc)<<RC<<1/(2pW)

18. Quadrature Modulation
Sends two info. signals on the cosine and sine carriers
DSBSC Demod
m1(t)
LPF
m1(t)cos(wct)+
m2(t)sin(wct)
cos(wct)
-90o
sin(wct)
m2(t)
DSBSC
Demod
LPF

19. Digital Communication System Block Diagram
Compressed Source Bits
Encoded Bits
Baseband
Waveform
m(t)
Modulated Waveform
s(t)
Bits
Digital Source
Error-Correction Encoding
Baseband
Modulation
Passband
Modulation
Bits
Compression
Analog Source
ADC
Removes redundancy
introduces controlled redundancy
binary or multi-level
shifts waveform
to carrier
frequency
converts
continuous-time to bits
propagates signal but adds distortion, noise
& interference
Channel
compares waveform to thresholds to detect bits
restores source redundancy
corrects errors in detected bits
shifts waveform to baseband
Bits
Digital Sink
Error-Correction Decoding
Baseband Demodulation
Passband
Demodulation
Decompression
Analog Sink
DAC
ŝ(t):
Corrupted Copy
of s(t)
Decoded Compressed Source Bits
Detected Encoded
Bits
Demodulated
Waveform
𝑚 (t)
converts
bits to continuous-time
Analog System
Channel is a physical entity (wire, cable, wireless channel, string)
Cannot send a complex signal over a physical channel: s(t) must be real
S(jw)=S*(-jw): s(t) real/even  S(jw) real/even; s(t) real/odd  S(jw) imaginary/odd
Often write s(t) in terms of in-phase/quadrature components: s(t)=sI(t)cos(wct)-sQ(t)sin(wct)

20. Baseband Digital Modulation
Baseband digital modulation converts bits into analog signals y(t) (bits encoded in amplitude)
Pulse shaping (optional topic)
Instead of the rect function, other pulse shapes used
Improves bandwidth properties and timing recovery
Explored in extra credit Matlab problem
Polar
On-Off A A
m(t) Tb
m(t) t t -A

21. Passband Digital Modulation
Changes amplitude (ASK), phase (PSK), or frequency (FSK, no covered) of carrier relative to bits
We use baseband digital modulation as information signal m(t) to encode bits, i.e. m(t) is on-off or polar
Passband digital modulation for ASK/PSK is a special case of DSBSC
For m(t) on/off (ASK) or polar (PSK), modulated signal is

22. ASK and PSK Amplitude Shift Keying (ASK) Phase Shift Keying (PSK) A -A A
m(t) -A
AM Modulation A
m(t) -A
AM Modulation
Assumes carrier phase f=0, otherwise need phase recovery of f in receiver

23. ASK/PSK Demodulation
Similar to AM demodulation, but only need to choose between one of two values (need coherent detection)
Decision device determines which of R0 or R1 that R(nTb) is closest to
For ASK, R0=0, R1=A, For PSK, R0=-A, R1=A
Noise immunity DN is half the distance between R0 and R1
Bit errors occur when noise exceeds this immunity
Decision Device
Integrator
(LPF)
nTb R1
r(nTb)
r(nTb)+N
“1” or “0”
s(t)  a0 DN
r(nTb) R0
cos(wct+f)

24. Quadrature Digital Modulation: MQAM
Sends different bit streams on the sine and cosine carriers
Baseband modulated signals can have L>2 levels
More levels for the same TX power leads to smaller noise immunity and hence higher error probability:
Sends 2log2(L)=M bits per symbol time Ts, Data rate is M/Ts bps
Called MQAM modulation: 10 Gbps WiFi: 1024-QAM (10 bits/10-9 secs)
L=32 levels
Decision Device
“1” or “0” R0 R1 a0
rI(nTb)
+NI
mi(t)
m1(t)cos(wct)+
m2(t)sin(wct)+
n(t) X
-90o
cos(wct)
sin(wct) A Ts
A/3 t
-A/3 -A
Decision Device
“1” or “0” R0 R1 a0
rQ(nTb)
+NQ
Data rate: log2L bits/Ts
Ts is called the symbol time

25. Introduction to FIR Filter Design
Signal processing today done digitally
Cheaper, more reliable, more energy-efficient, smaller
Discrete time filters in practice must have a finite impulse response: h[n]=0, |n|>M/2
Otherwise processing takes infinite time
FIR filter design typically entails approximating an ideal (IIR) filter with an FIR filter
Ideal filters include low-pass, bandpass, high-pass
Might also use to approximate continuous-time filter
We focus on two approximation methods
Impulse response and filter response matching
Both lead to the same filter design

26. Impulse Response Matching
Given a desired (noncausal, IIR) filter response hd[n]
Objective: Find FIR approximation ha[n]: ha[n]=0 for |n|>M/2 to minimize error of time impulse response
By inspection, optimal (noncausal) approximation is
Doesn’t depend on ha[n] W ( ) j a e H p -
Exhibits Gibbs phenomenon
from sharp time-windowing

27. Frequency Response Matching
Given a desired frequency response Hd(ejW)
Objective: Find FIR approximation ha[n]: ha[n]=0 for |n|>M/2 that minimizes error of freq. response
Set and
By Parseval’s identity:
Time-domain error and frequency-domain error equal
Optimal filter same as in impulse response matching
and

28. Causal Design and Group Delay
Can make ha[n] causal by adding delay of M/2
Leads to causal FIR filter design
If Ha(ejW) constant, H(ejW) linear in W with slope -.5M
Most filter implementations do not have linear phase, corresponding to a constant delay for all W.
Group delay defined as
Constant for linear phase filters
Piecewise constant for piecewise linear phase filters
Nonconstant group delay introduces phase distortion relative to an ideal filter

29. Art and Science of Windowing
Window design is created as an alternative to the sharp time-windowing in ha[n]
Used to mitigate Gibbs phenomenon
Window function (w[n]=0, |n|>M/2) given by
Windowed noncausal FIR design:
Frequency response smooths Gibbs in Ha(ejW)
Design often trades “wiggles” in main vs. sidelobes
Hamming smooths out wiggles from rectangular window
Introduces more distortion at transition frequencies than rectangle

30. Typical Window Designs
0.5 1
1.5 2
2.5 3
-0.2
0.2
0.4
0.6
0.8 W ( e j ) M
= 16
Boxcar
Triangular
Hamming
Hanning

31. Summary of FIR Design
We are given a desired response hd[n] which is generally noncausal and IIR
Examples are ideal low-pass, bandpass, highpass filters
May be derived from a continuous-time filter
Choose a filter duration M+1 for M even
Larger M entails more complexity/delay, less approximation error e
Design a length M+1 window function w[n], real and even, to mitigate Gibbs while keeping good approximation to hd[n]
Calculate the noncausal FIR approximation ha[n]
Calculate the noncausal windowed FIR approximation hw[n]
Add delay of M/2 to hw[n] to get the causal FIR filter h[n]

32. FIR Realization: Direct Form
Consists of M delay elements and M+1 multipliers
Can introduce different delays at different freq. components of x[n]
Will discuss more when we cover z transforms
Efficient implementation using Discrete-Fourier Transform (DFT)

33. Main Points
Sampling and reconstruction bridges analog and digital worlds
Upsampling and downsampling ease implementation requirements
Analog and digital communications allows transmission of information signals through the airwaves
FIR filter design approximates perfect filters with a design tailored to a set of engineering tradeoffs.