4-FilteringPart2

Digital Filtering Part 2:
Pulse Shaping
ELEC 433 - Spring 2013
Evan Everett and Michael Wu

Output of modulator
[01]
[11]
[00]
[10]

Output of modulator
[01]
[11]
[00]
[10]
Train of symbols
1 0

Output of modulator
[01]
[11]
[00]
[10]
Train of symbols
1 0

A single symbol in the time domain
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
Time
5 10 15 20 25 30 35 40 45 50

A single symbol in the frequency domain
·10 −2
3
2
1
0
−1
Frequency
• A sinc in frequency → infinite bandwidth
• FCC angry!
• Band-limited RF/antennas

A single symbol in the frequency domain
·10 −2
3
2
1
0
X
X
−1
Frequency
What if we only keep a small piece?

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
What if we only keep a small piece?
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
0.2
0
−0.2
5 10 15 20 25 30 35 40 45 50
Time
5 10 15 20 25 30 35 40 45 50
The downside: our signal is “wider” in time

A sequence of symbols
1.8
1.6
1.4
1.2
1
0.8
0.6
Symbol 1
0.4
0.2
1.8
1.6
1.4
1.2
1
0.8
0.6
Symbol 2
0.4
0.2
Symbol 3
0
0
Time
−0.2
−0.2
10 15 20 25 30 5 10 35 15 40 20 45 25 50 30 5 10 35 40 15 45 20 25 50 30 35 40

A band-limited sequence of symbols
1.8
1.6
1.4
1.2
1
0.8
1.8
1.6
1.4
1.2
1
0.8
0.6
0.6
Symbol 1 Symbol 2 Symbol 3
0.4
0.4
0.2
0
0.2
0
−0.2
Time
−0.2
5 10 15 20 25 305 35 10 40 15 45 20 50 25 530 1035 1540 2045 2550
30 35 40 45
Band-limiting caused
Inter-Symbol Interference (ISI)

Pulse Shaping Requirements
• Frequency domain: low-pass response
• H(ω)≈1 in passband, H(ω)≈0 in stopband
• Time domain: zero ISI and finite extent
• Impulse response needs periodic zeros
• Zeros must occur at other data sample times
• h[nτ] = 0 where τ is data sample period

Zero-ISI Example
t
-3T
-2T
-T
T 2T 3T

Raised Cosine Function
• Most common pulse shaping filter
• Meets both key requirements
• Easy to implement digitally
• Impulse response:
x(t) = sin πt cos παt
τ
τ
2
πt
τ
1 − 2αt
τ
= sinc t
τ
cos
παt
1 − 2αt
τ
τ
2
τ
α
: Sampling period- determines zero crossings of x(t)
: Rolloff factor- determines passband (“excess bandwidth”)

Raised Cosine Function
Impulse Response
0.8
0.6
=0
=.15
=.3
=.5
=1
0.4
0.2
0
-0.2
-3 -2 -1 0 1 2 3

Raised Cosine Function
Properties of the Impulse Response
0.8
0.6
=0
=.15
=.3
=.5
=1
0.4
0.2
0
-0.2
-3 -2 -1 0 1 2 3
x(0) = 1 for all α

Raised Cosine Function
Properties of the Impulse Response
0.8
0.6
=0
=.15
=.3
=.5
=1
0.4
0.2
0
-0.2
-3 -2 -1 0 1 2 3
Zeros do not depend on α

Raised Cosine Function
Properties of the Impulse Response
0.8
0.6
=0
=.15
=.3
=.5
=1
0.4
0.2
0
-0.2
-3 -2 -1 0 1 2 3
Size of ripples depend on α

Raised Cosine Function
Properties of the Impulse Response
0.8
0.6
=0
=.15
=.3
=.5
=1
0.4
0.2
0
-0.2
-3 -2 -1 0 1 2 3
α=0 gives sinc(x)

Raised Cosine Function
Frequency Response:
X(ω) =
( )
⎧⎧
⎪⎪τ, ω ≤ π 1− α
τ
⎪⎪
⎪⎪τ
⎡⎡
2 1− sin ⎛⎛ ⎛⎛ τ ⎞⎞
⎝⎝
⎜⎜
2α ⎠⎠
⎟⎟ ⎛⎛
ω − π ⎞⎞ ⎞⎞ ⎤⎤
⎨⎨ ⎢⎢
⎝⎝
⎜⎜ ⎝⎝
⎜⎜
τ ⎠⎠
⎟⎟
⎠⎠
⎟⎟
⎣⎣
⎦⎦
⎥⎥ , π ( 1 − α)
⎪⎪
τ
⎪⎪
⎪⎪0, ω ≥ π 1+ α
⎩⎩
τ
< ω < π ( 1 + α)
τ
( )
τ : Sampling period- determines zero crossings of x(t)
α: Rolloff factor- determines passband (“excess bandwidth”)

Raised Cosine Function
1
0.9
0.8
0.7
Frequency Response
=0
=.15
=.3
=.5
=1
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.5 1 1.5 2

Raised Cosine Function
Properties of the Frequency Response
1
0.9
0.8
0.7
=0
=.15
=.3
=.5
=1
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.5 1 1.5 2
α=0 gives notch filter

Raised Cosine Function
Properties of the Frequency Response
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(1- α)
(1+ α)
=0
=.15
=.3
=.5
=1
0
0 0.5 1 1.5 2
α sets the filter’s passband

Raised Cosine Function
Properties of the Frequency Response
1
0.9
0.8
0.7
=0
=.15
=.3
=.5
=1
0.6
0.5
0.4
X(ω(1+α))=0
0.3
0.2
0.1
0
0 0.5 1 1.5 2

Raised Cosine Function
Properties of the Frequency Response
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
X(ω(1+α))=0 (1+ 0
0 0.5 1 1.5 2
α=0 α gives sets the notch filter’s passband

Square Root Raised Cosine
H RC (ω)
H RRC (ω) = H RC (ω)
1
1
0.9
=0
=.15
0.9
=0
=.15
0.8
=.3
=.5
0.8
=.3
=.5
0.7
=1
0.7
=1
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0 0.5 1 1.5 2
0
0 0.5 1 1.5 2
Why is this useful?
• Split pulse shaping filter between Tx and Rx
• Pulse shaping becomes matched filtering

Matched Filtering with RRC
TX:
Prevent out-of-band
interference
H RRC (ω)
RX:
Reject out-of-band
interference/noise
H RRC (ω)
Data
Mod
∼ ∼
Matched RRC Filters
H(ω) = H RRC (ω)H RRC (ω)
= H RC (ω) H RC (ω)
Data
Demod
Net response is
zero ISI, unity at
sample times!
= H RC (ω) (1

Raised Cosine Filtering
Adapted from MATLAB’s ‘playshow rcosdemo’
1.5
1
0.5
Amplitude
0
-0.5
-1
-1.5
0 5 10 15 20 25 30
Time
Start with a binary data stream

Raised Cosine Filtering
Adapted from MATLAB’s ‘playshow rcosdemo’
1.5
1
0.5
Amplitude
0
-0.5
-1
-1.5
0 5 10 15 20 25 30
Time
Apply the raised cosine filter to the data

Raised Cosine Filtering
Adapted from MATLAB’s ‘playshow rcosdemo’
1.5
1
0.5
Amplitude
0
-0.5
-1
-1.5
0 5 10 15 20 25 30
Time
Delay the data to see the preservation of the information

Raised Cosine Filtering
Adapted from MATLAB’s ‘playshow rcosdemo’
1.5
1
α=0.2
α=0.5
0.5
Amplitude
0
-0.5
-1
-1.5
0 5 10 15 20 25 30
Time
See the effect of changing α

Raised Cosine Filtering
Adapted from MATLAB’s ‘playshow rcosdemo’
1.5
1
0.5
Amplitude
0
-0.5
-1
-1.5
0 5 10 15 20 25 30
Time
Using a square root raised cosine filter instead

Raised Cosine Filtering
Adapted from MATLAB’s ‘playshow rcosdemo’
1.5
1
0.5
Amplitude
0
-0.5
-1
-1.5
0 5 10 15 20 25 30
Time
Data reconstructed using root-raised filter again

Digital Pulse-shaping: Tx
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
Amplitude
0
Amplitude
0
Amplitude
0
−0.5
−0.5
−0.5
−1
−1
−1
Data
−1.5
−1.5
−1.5
0 5 10 15
0
20
5
25
10
30
15 20 25 30
0 5 10 15
Time
Time
Tim
Modulator Interpolating Filter
∼
D/A Tx

1.5
1.5
Digital Pulse-shaping: Rx
1.5
1
1
1
0.5
0.5
0.5
0
Amplitude
0
Amplitude
0
0.5
−0.5
−0.5
−1
−1
−1
−1.5
1.5
−1.5
0 5 10 15 20 25 30
0 5 10 15 20 25 30
Time 0 5 10 15 20 25 30
Rx A/D
Time
∼
Time
Data
Decimating Filter
Demodulator

Interpolating Filters
• Commonly required in DSP/Comm
• DSP: smoothing, softening pixelation
• Comm: upsampling without aliasing, pulse shaping
• Example:
Original
Enlarged w/o
interpolation
Interpolated

Implementing Multi-rate Filters
Interpolation
M
N-tap LPF
• Example: interpolate by a factor of M = 8
• Low-pass filter of length N = 32
• Suppose we upsample (zero pad) and then use a serial filter
• How much faster does the internal rate have to be than input
rate?
• M*N = 8*32 = 256x as fast
• If the input rate is near the speed of our FPGA, this is hopeless

Implementing Multi-rate Filters
• Basic multi-rate filters are very inefficient
• Inputs are mostly zeros after upsampling
• Outputs are mostly discarded after downsampling
• Smarter design can save a lot of resources
Interpolation
M
Mostly Zeros
N-tap LPF
Decimation
Mostly Discarded
N-tap LPF
M

4x Interpolation Example
x0 x1 x2 …
M
M=4
N-tap LPF
Impulse Response

Impulse Response
Data
4x Interpolation Example
x 0 0 0 0 x 1 0 0 0 x 2 0 0 0 x 3 0 0 0

Impulse Response
4x Interpolation Example
Data
x 0 0 0 0 x 1 0 0 0 x 2 0 0 0 x 3 0 0 0
0 x 0 0 0 0 x 1 0 0 0 x 2 0 0 0 x 3 0 0

Impulse Response
4x Interpolation Example
Data
x 0 0 0 0 x 1 0 0 0 x 2 0 0 0 x 3 0 0 0
0 x 0 0 0 0 x 1 0 0 0 x 2 0 0 0 x 3 0 0
0 0 x 0 0 0 0 x 1 0 0 0 x 2 0 0 0 x 3 0

Impulse Response
4x Interpolation Example
Data
x 0 0 0 0 x 1 0 0 0 x 2 0 0 0 x 3 0 0 0
0 x 0 0 0 0 x 1 0 0 0 x 2 0 0 0 x 3 0 0
0 0 x 0 0 0 0 x 1 0 0 0 x 2 0 0 0 x 3 0
0 0 0 x 0 0 0 0 x 1 0 0 0 x 2 0 0 0 x 3

Polyphase Filters
• Break impulse response into M sub-filters
• h[0...N-1] is broken down as:
• h0=h[0,M,2M,3M,...]
• h1=h[1,M+1,2M+1,3M+1,...]
• h2=h[2,M+2,2M+2,3M+2,...]
• ... and so on

Parallel Polyphase Filter
• Then cycle through sub-filter outputs at rate M
• Output is the same as basic version
• Only the multiplexer runs at rate M
h 0
h 1
h 2
...
h M-1

Serial Polyphase Filters
• Replace subfilters with clever ROM/RAM indexing
• Most efficient implementation: no unused logic
• Core runs at rate N*M/M (for original h[0...N-1])
Data
Addressable
Shift Register
X +
Counter
Branch Index
+
Coefficient
ROM z -1
Serial Interpolating Polyphase Filter