EE445S Real-Time Digital Signal Processing Lab
Spring 2016
Interpolation and Pulse Shaping
Prof. Brian L. Evans
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
Lecture 7
http://www.ece.utexas.edu/~bevans/courses/realtime
Outline
• Discrete-to-continuous conversion
• Interpolation
• Pulse shapes
Rectangular
Triangular
Sinc
Raised cosine
• Sampling and interpolation demonstration
• Conclusion
7-2
Data Conversion
Data Conversion
• Analog-to-Digital Conversion
Lowpass filter has
stopband frequency
less than ½ fs to reduce
aliasing at sampler output
(enforce sampling theorem)
Lecture 4
Analog
Lowpass
Filter
• Digital-to-Analog Conversion
Discrete-to-continuous
conversion could be as
simple as sample and hold
Lowpass filter has stopband
frequency less than ½ fs to
reduce artificial high frequencies
Lecture 8
Quantizer
Sampler at
sampling
rate of fs
Lecture 7
Discrete to
Continuous
Conversion
Analog
Lowpass
Filter
fs
7-3
Discrete-to-Continuous Conversion
• Input: sequence of samples y[n]
• Output: smooth continuous-time function obtained
through interpolation (by “connecting the dots”)
If f0 < ½ fs , then
y[n]
y[n]  A cos2  f 0 Ts n   
3
4
5
6
7
n
would be converted to
1
2
~
y (t )
~
y (t )  A cos2  f 0 t   
Otherwise, aliasing has occurred, and the converter would
reconstruct a cosine wave whose frequency is equal to the
aliased positive frequency that is less than ½ fs
7-4
Discrete-to-Continuous Conversion
• General form of interpolation is sum of weighted

pulses
~
y (t ) 
 y[n] p(t  T
n  
s
n)
Sequence y[n] converted into continuous-time signal that is an
approximation of y(t)
Pulse function p(t) could be rectangular, triangular, parabolic,
sinc, truncated sinc, raised cosine, etc.
Pulses overlap in time domain when pulse duration is greater
than or equal to sampling period Ts
Pulses generally have unit amplitude and/or unit area
Above formula is related to discrete-time convolution
7-5
Interpolation From Tables
• Using mathematical tables of
numeric values of functions to
compute a value of the function
• Estimate f(1.5) from table
Zero-order hold: take value to be f(1)
to make f(1.5) = 1.0 (“stairsteps”)
Linear interpolation: average values of
nearest two neighbors to get f(1.5) = 2.5
Curve fitting: fit four points in table to
polynomal a0 + a1 x + a2 x2 + a3 x3
which gives f(1.5) = x2 = 2.25
x
f(x)
0
1
2
3
0.0
1.0
4.0
9.0
~
f ( x)
9
4
1
x
0
1
2
3
7-6
Rectangular Pulse
• Zero-order hold
Easy to implement in hardware or software
 t  1 if  1 Ts  t  1 Ts
p(t )  rect    
2
2
T
 s  0
otherwise
The Fourier transform is
P( f )  Ts sinc  f Ts   Ts
p(t)
1
-½ Ts
½ Ts
t
sin(  f Ts )
sin  x 
where sinc ( x) 
 f Ts
x
In time domain, no overlap between p(t) and adjacent pulses
p(t - Ts) and p(t + Ts)
In frequency domain, sinc has infinite two-sided extent; hence,
the spectrum is not bandlimited
7-7
Sinc Function
sinc(x) 1
x
3 2

0 
2
3
sin x 
sinc x  
x
How to compute sinc(0)?
As x  0, numerator and
denominato r are both going
to 0. How to handle it?
Even function (symmetric at origin)
Zero crossings at x   ,  2 ,  3 , ...
Amplitude decreases proportionally to 1/x
7-8
Triangular Pulse
• Linear interpolation
It is relatively easy to implement in hardware or software,
p(t)
although not as easy as zero-order hold
 |t |
 t  1 
if Ts  t  Ts


p(t )      Ts
 Ts  
otherwise
 0
1
-Ts
Ts
t
Overlap between p(t) and its adjacent pulses p(t - Ts) and
p(t + Ts) but with no others
• Fourier transform is
P( f )  Ts sinc 2  f Ts 
How to compute this? Hint: Triangular pulse is equal to 1 / Ts
times the convolution of rectangular pulse with itself
In frequency domain, sinc2(f Ts) has infinite two-sided extent;
hence, the spectrum is not bandlimited
7-9
Sinc Pulse
• Ideal bandlimited interpolation

sin 
T
 
p (t )  sinc  t    s

 Ts 
t
Ts

t 

f 
1
P( f )  rect  
Ts
 Ts 
W
1
2 Ts
In time domain, infinite overlap between other pulses
Fourier transform has extent f  [-W, W], where
P(f) is ideal lowpass frequency response with bandwidth W
In frequency domain, sinc pulse is bandlimited
• Interpolate with infinite extent pulse in time?
Truncate sinc pulse by multiplying it by rectangular pulse
Causes smearing in frequency domain (multiplication in time
domain is convolution in frequency domain)
7 - 10
Raised Cosine Pulse: Time Domain
• Pulse shaping used in communication systems
 t  cos2  W t 
p(t )  sinc  
 T  1  16  2 W 2 t 2
 s
ideal lowpass filter Attenuation by 1/t2 for
impulse response large t to reduce tail
W is bandwidth of an
ideal lowpass response
  [0, 1] rolloff factor
Zero crossings at
t =  Ts ,  2 Ts , …
Simon Haykin, Communication Systems, 3rd ed.
• See handout G in reader on raised cosine pulse
7 - 11
Raised Cosine Pulse Spectra
• Pulse shaping used in communication systems
Bandwidth increased
by factor of (1 + ):
(1 + ) W = 2 W – f1
f1 marks transition from
passband to stopband
Simon Haykin, Communication Systems, 3rd
ed.
1
1

if 0  | f | f1

2W

  | f |  W   
 1 
  if f1  | f |  2W  f1
P( f )  
1  sin 

 2 W  2 f1  
 4W 
0
otherwise


W
2 Ts
  1
Bandwidth generally scarce in communication systems
f1
W
7 - 12
Sampling and Interpolation Demo
• DSP First, Ch. 4, Sampling and interpolation
http://www.ece.gatech.edu/research/DSP/DSPFirstCD/
Sample sinusoid y(t) to form y[n]
Reconstruct sinusoid using
rectangular, triangular, or
truncated sinc pulse p(t)
~
y (t ) 

 y[n] p(t  T
n  
s
n)
• Which pulse gives the best reconstruction?
• Sinc pulse is truncated to be four sampling periods
long. Why is the sinc pulse truncated?
• What happens as the sampling rate is increased?
7 - 13
Conclusion
• Discrete-to-continuous time conversion involves
interpolating between known discrete-time samples
y[n]
y[n] using pulse shape p(t)
~
y (t ) 

 y[n] p(t  T
n  
s
n)
• Common pulse shapes
3
4
5
6
7
n
1
2
~
y (t )
Rectangular for same-and-hold interpolation
Triangular for linear interpolation
Sinc for optimal bandlimited linear interpolation but impractical
Truncated raised cosine for practical bandlimited interpolation
• Truncation in time domain causes smearing in
7 - 14
frequency domain