1. Lecture 3 Outline: Sampling and Reconstruction
Announcements:
Discussion: Monday 7-8 PM, Location (likely) Hewlett 102
TA OHs (will be held in kitchen area outside Packard 340):
Alon 6-7 PM on Monday
Mainak 4-5 PM on Tuesday, 1-2 PM on Wednesday
Jeremy 3-4 PM on Friday, 3-4 PM on Tuesday.
First HW posted, due next Wed. 5pm
TGIF treats
Finish “Fun with Fourier”
Sampling
Reconstruction
Nyquist Sampling Theorem

2. Review of Last Lecture Duality Relationships:
Connections between Continuous/Discrete Time
Discrete time less intuitive than continuous time (for most people)
Can consider a discrete time process as samples of a continuous time process, which is periodic in frequency
Filtering (clarification to Case 2: W=2p/T=6p/T0)
.5T
-.5T A t
x(t) AT w
X(jw) AT t
x(t)
.5W
-.5W AW
w=2pf
X(jw)
Mathematically Sound
Definition
LPF: X(±jW)=.5
Similarly for P(t) a2 a1 a0 a3 w a4
a-2
a-3
a-4 a5 a6
a-5
a-6
a-1
Include a3,a-3? T0
-T0
.5T
-.5T …
X(jw)
Depends on value of X(jw) at w=W 1
0.5
If X(±jW)>0, yes, if zero then no
Assumed last lecture -W W W

3. Filtering and Convolution
Filtering Example: Discrete Periodic Pulse
Important convolution examples
OWN pg. 392 2W a0
x[n]: Periodic Discrete
X(ejW)
a-1 a1
a-4 W -W 1 a4 -N
-N1 N1 N n
a-3 a3
a-2 a2
NA2
3A2
3A2 2T
A2T
2A2
2A2 T A T A A * = * = A2 A2 … … … …
N-1
N-1
N-1
2N-1

4. Periodic Signals Continuous time: Discrete time: … … … …
x(t) periodic iff there exists a T0>0 such that x(t)=x(t+T0) for all t
T0 is called a period of x(t); smallest such T0 is fundamental period
If x(t) periodic with period T0, y(t) periodic with period T1, then x(t)+y(t) is periodic with period k0T0=k1T1 if such integers ki exist.
Discrete time:
x[n] periodic iff there exists a N0>0 such that x[n]=x[n+N0] for all n
N0 is called a period of x(t); smallest such N0 is fundamental period
If x[n] periodic with period N1, y[n] periodic with period N2, then x[n]+y[n] is periodic with period k1N1=k2N2 if such integers ki exist. T0
2T0
-T0
.5T
-.5T t
k0T0 T0 …
2T0
k1T1 T1 …
2T1
-N0 -N N N0 n …
k1N1 N1 …
2N1
2N0
k0N0 N0

5. Energy, Power, and Parseval’s Relation
Energy signals have zero power; Power signals have infinite energy
Continuous Time Aperiodic Signals
Discrete Time Aperiodic Signals
−∞ ∞ |𝑥 𝑡 | 2 𝑑𝑡= 1 2𝜋 −∞ ∞ |𝑋 𝑗𝜔 | 2 𝑑𝜔
Periodic:
𝑛=−∞ ∞ |𝑥 𝑛 | 2 = 1 2𝜋 2𝜋 |𝑋 𝑒 𝑗 | 2 𝑑
Periodic:

6. Examples Continuous-time Discrete-time
x(t)
X(jw) AT w
Which would you rather integrate? a0 a1 a2 a3 a4
a-2
a-1
a-3
a-4 N 2N -N N1
-N1 n
x[n]: Periodic Discrete
Which would you rather sum?

7. Sampling and Reconstruction vs
Sampling and Reconstruction vs. Analog-to-Digital and Digital-to-Analog Conversion
Sampling: converts a continuous-time signal to a 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

8. Applications Capture: audio, images, video
Storage: CD, DVD, Blu-Ray, MP3, JPEG, MPEG
Signal processing: compression, enhancement and synthesis of audio, images, video
Communication: optical fiber, cell phones, wireless local-area networks (WiFi), Bluetooth
Applications: VoIP, streaming music and video, control systems, Fitbit, Occulus Rift

9. Sampling Sampling (Time): = Sampling (Frequency) = * x(t) nd(t-nTs)
xs(t) Ts
2Ts
3Ts
4Ts
-3Ts
-2Ts
-Ts
-3Ts
-2Ts
-Ts Ts
2Ts
3Ts
4Ts =
X(jw) *
nd(t-n/Ts)
Xs(jw)
-2p Ts 2p Ts
-2p Ts 2p Ts

10. Reconstruction Frequency Domain: low-pass filter
Time Domain: sinc interpolation
H(jw)
Xs(jw)
H(jw)
-2p Ts
Xs(jw) 2p
Xr(jw) 1 -W W w w
-2p Ts 2p Ts

11. 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

12. Main Points
Sum of periodic signals is periodic if integer multiples of each period are equal
Energy and power can be computed in time or frequency domain
Sampling bridges the analog and digital worlds, with widespread applications in the capture, storage, and processing of signals
Sampling converts continuous-time signals to sampled signals
Multiplication with delta train in time, convolution with delta train in frequency
Reconstruction recreates a continuous-time signal from its samples
Multiplication with LPF in frequency, sinc interpolation in time
A bandlimited signal of bandwidth W sampled at or above its Nyquist rate of 2W can be perfectly reconstructed from its samples