Lecture 3 Outline:
Sampling and Reconstruction
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Announcements:
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Discussion: Monday 7-8 PM, Location (likely) Hewlett 102
TA OHs (will be held in kitchen area outside Packard 340):
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
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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
Review of Last Lecture

Duality Relationships: x(t )  X ( jw )  X (t )  2x(  jw )
x[n ]  {ak } 
x(t)
A
.5T
-.5T

AT
X(jw) AT
t
 2
T
2
T
AW
AT
x(t)
w
1
x[  n ]
N
a[ k ]  a k 
 2
W
X(jw)
2A
.5W w2f
-.5W
t
2
W
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
Mathematically Sound
process, which is periodic in frequency
 Filtering
Include a3,a-3?
Definition
a0
LPF: X(±jW)=.5
Similarly for P(t)
(clarification to Case 2: W=2/T=6/T0)
X(jw)
…
…
Depends on value of X(jw) at w=W
-.5T .5T
-T0
0
0
If X(±jW)>0,
yes,
if zeroTthen
no
-W
Assumed last lecture
W
1
0.5
0
W
a-6
a-4
a-5
a-3
a-1
a1
 6  2 0 2
T0 a-2 T
T0
0
a3
6
a2 T
0
a4
a5
a6
w
Filtering and Convolution

Filtering Example: Discrete Periodic Pulse
OWN pg. 392
2W
a0
x[n]: Periodic Discrete
X(ejW)
a-1
a-4
1
-N

-N1 0 N1
-W
n
N
W
a1
W
a4
  a-3
Important convolution examples
a-2
 2
N
2
N

a2 a3
NA2
3A2
A2T
3A2
2A2
A
A
=
*
0
T
A
0
T
…
0
2T
0
N-1
*
…
0
N-1
=
2A2
A2
…
0
A2
…
N-1
2N-1
Periodic Signals

Continuous time:



0 T0


2T0 t
…
k0T0
2T0
0 T1 2T1
k1T1
Discrete time:

T0
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.
…

-.5T0 .5T
-T0
-N0
-N 0 N
N0
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.
…
…
0 N0
2N0
k0N0
0 N1 2N1
k1N1
n
Energy, Power, and
Parseval’s Relation
Energy signals have zero power; Power signals have infinite energy

Continuous Time Aperiodic Signals
∞
1
2
|𝑥 𝑡 | 𝑑𝑡 =
2𝜋
−∞
∞
|𝑋 𝑗𝜔 |2 𝑑𝜔
−∞
Periodic:

Discrete Time Aperiodic Signals
∞
|𝑥 𝑛
|2
𝑛=−∞
Periodic:
1
=
2𝜋
|𝑋 𝑒 𝑗 W |2 𝑑W
2𝜋
Examples

Continuous-time
A
-.5T

E   | x (t ) | dt 
2


.5T
 2
T
t
2
T
w
AP (t / T )  ATsinc.5wT /  
.5T
 A dt  A T
2
X(jw) AT
x(t)
2
.5T
Which would you rather integrate?
Discrete-time
a0
x[n]: Periodic Discrete
a-1
-N
E
1
N
-N1 0 N1
 | x[n] |2 
n  N 
N
1
N
2N
n
1  ( 2 N
n   N1
a-4
  a-3
N1
1
 1) / N
a1
(2 N  1) / N
k  iN

 sin(2k /1N ) /( N  .5)
ak  
1
else

N sin(k / N )
a4
a-2
 2
N0
2
N0

a2 a3
Which would you rather sum?
Sampling and Reconstruction vs. Analog-toDigital and Digital-to-Analog Conversion

Sampling: converts a continuous-time signal to a sampled
signal

-3Ts -2Ts -Ts 0
Ts 2Ts 3Ts 4Ts
Reconstruction: converts a sampled signal to a continuoustime signal.
-3Ts -2Ts -Ts 0

Ts 2Ts 3Ts 4Ts
Analog-to-digital conversion: converts a continuous-time
signal to a discrete-time quantized or unquantized signal
Each level can
be represented
by 0s and 1s

-3 -2 -1 0
1 2
3
4
Digital-to-analog conversion. Converts a discrete-time
quantized or unquantized signal to a continuous-time signal.
-3 -2 -1 0
1 2
3
4
Applications
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Capture: audio, images, video
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Storage: CD, DVD, Blu-Ray, MP3, JPEG,

Signal processing: compression, enhancement
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Communication: optical fiber, cell phones,

Applications: VoIP, streaming music and video,
MPEG
and synthesis of audio, images, video
wireless local-area networks (WiFi), Bluetooth
control systems, Fitbit, Occulus Rift
Sampling

Sampling (Time):
nd(t-nTs)
x(t)
-3Ts -2Ts -Ts 0
0

=
xs(t)
-3Ts -2Ts -Ts 0
Ts 2Ts 3Ts 4Ts
Ts 2Ts 3Ts 4Ts
Sampling (Frequency)
X(jw)
0
*
2
Ts
-2
Ts
nd(t-n/Ts)
0
=
2
Ts
Xs(jw)
-2
Ts
0
2
Ts
Reconstruction

Frequency Domain: low-pass filter
Xs(jw)
Xr(jw) H(jw)
H(jw)
Xs(jw)
1
-W
-2
Ts

0
W
w
-2
Ts
2
Ts
0
2
Ts
w
Time Domain: sinc interpolation
 t  nTs 

xr t    xs ( nTs )h t  nTs    xs ( nTs ) sinc
T
n  
n  
s




1
X r  jw   H ( jw ) 
 Ts


X


k  



2
j  w  k
Ts

 
 

 
Nyquist Sampling Theorem

A bandlimited signal [-W,W] radians is completely described
by samples every Ts/W secs.

The minimum sampling rate for perfect reconstruction, called
the Nyquist rate, is W/ samples/second

If a bandlimited signal is sampled below its Nyquist rate,
distortion (aliasing occurs)
X(jw)
-W
0
W
-2W W
X(jw)
Xs(jw)
0
2W=2/Ts
W
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


Reconstruction recreates a continuous-time signal from its samples


Multiplication with delta train in time, convolution with delta train in frequency
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