DSP Group, EE, Caltech, Pasadena CA
IIR Ultra-Wideband Pulse Shaper Design
Chun-yang Chen and P.P. Vaidyananthan
California Institute of Technology
DSP Group, EE, Caltech, Pasadena CA
The UWB communications
In 2002, the Federal Communication Community (FCC) approved a spectral mask for operation of UWB devices.
It allows UWB devices operate on 3.1GHz ~ 10.6GHz under -41.3dBm.
DSP Group, EE, Caltech, Pasadena CA
Impulse radio system for UWB
Impulse radio system transmits very short pulses p(t) without RF carriers.
The radiated power spectrum of impulse radio system can be expressed by
22)()()()( fPfSfHfS meq )( fM
Fourier transform of
the pulse
Depends on the modulation method
Transfer function from modulated
pulse train to radiated signal
DSP Group, EE, Caltech, Pasadena CA
Example of Gaussian monocycle pulse
For example, if we use the Gaussian monocycle pulse (derivative of a Gaussian pulse), then
Assume
2)(2
2)( g
t
g
et
etp
1)()( fSfH meq
Then the radiated power spectrum is2222
)2
exp()( gfffS
DSP Group, EE, Caltech, Pasadena CA
Example of Gaussian monocycle pulse (2)
The power spectrum for using Gaussian monocycle pulse
0 2 4 6 8 10 12-100
-90
-80
-70
-60
-50
-40
Pow
er(d
Bm
)
Frequency(GHz)
Maskg=10ns
g=50ns
g=100ns
The transmitting power is very
small.
2222
)2
exp()( gfffS
DSP Group, EE, Caltech, Pasadena CA
The optimization problem
To utilize the bandwidth, the optimal pulse should be designed so that the transmitting power is maximized.
)()()()( subject to
)()()(max
22
22
)(
max
min
fMfPfSfH
dffPfSfH
meq
F
F
meqtp
)()()()( 22
fMfPfSfH meq
The ideal solution to this problem is the pulse such that
DSP Group, EE, Caltech, Pasadena CA
Mask filling efficiency
The mask filling efficiency [Lewis et al. 2004] is defined as
max
min
max
min
)(
)()()(22
F
F
F
F meq
dffM
dffPfSfH
)()()()( 22
fMfPfSfH meq
The ideal solution
yields 100% of efficiency.
DSP Group, EE, Caltech, Pasadena CA
Pulse shaper
However, we cannot generate pulse with arbitrary with analog circuits.)( fP
We can generate the pulse by shaping the available waveforms by
M
nn nTtgbtp
0
)()(
This waveform can be directly generated by
analog circuit.
DSP Group, EE, Caltech, Pasadena CA
The scheme of FIR pulse shaper
D denotes the analog delay.
M
nn nTtgbtp
0
)()(
DSP Group, EE, Caltech, Pasadena CA
Power spectrum of the radiated signal
The Fourier transform of the pulse is
.)()(0
2
M
n
fT
ni
n fGebfP
M
nn nTtgbtp
0
)()(
).()()()(22
2
0
2
fSfHfGebfS meq
M
n
fT
ni
n
The power spectrum of the radiated signal is
DSP Group, EE, Caltech, Pasadena CA
Design of the pulse shaper To approximate the ideal solution, we choose
the shaper so that
).()()()()(22
2
0
2
fMfSfHfGebfS meq
M
n
fT
ni
n
}{ nb
It reduces to an FIR filter design problem. Standard technique such as the Parks-McClella
n algorithm can be used to design such a filter [Luo et all. 2003].
DSP Group, EE, Caltech, Pasadena CA
Results of using the pulse shaper
0 2 4 6 8 10 12-80
-75
-70
-65
-60
-55
-50
-45
-40
Frequency (GHz)
Pow
er (
dBm
)
The multipliers of the shaper is 17.
Gaussian monocycle
pulse
Gaussian monocycle pulse shaped by the minimax FIR filt
er
M
nn nTtgbtp
0
)()(
DSP Group, EE, Caltech, Pasadena CA
IIR pulse shaper
With the same complexity, IIR filters has better frequency response than FIR filters.
We can generate the pulse by summing the delay version of the elementary waveforms and the feedback
N
nn
M
nn nTtpanTtgbtp
10
)()()(
DSP Group, EE, Caltech, Pasadena CA
The scheme of IIR pulse shaper
D denotes the analog delay.
N
nn
M
nn nTtpanTtgbtp
10
)()()(
DSP Group, EE, Caltech, Pasadena CA
Power spectrum of the radiated signal
The Fourier transform of the pulse is
).()(
0
20
2
fG
ea
ebfP
M
n
fT
ni
n
M
n
fT
ni
n
N
nn
M
nn nTtpanTtgbtp
10
)()()(
).()()()(22
2
0
20
2
fSfHfG
ea
ebfS meqN
n
fT
ni
n
M
n
fT
ni
n
The power spectrum of the radiated signal is
DSP Group, EE, Caltech, Pasadena CA
Design of the IIR pulse shaper To approximate the ideal solution, we choose the
shaper and so that
).()()()()(22
2
0
20
2
fMfSfHfG
ea
ebfS meqN
n
fT
ni
n
M
n
fT
ni
n
Mnnb 0}{
It reduces to an IIR filter design problem.
Nnna 1}{
However, there is no standard technique to design IIR filter to fit arbitrary magnitude response.
DSP Group, EE, Caltech, Pasadena CA
Design of IIR pulse shaper using Elliptic filters
There are standard techniques to design IIR filters to fit bandpass magnitude responses such as elliptic IIR filters.
0 2 4 6 8 10 12-80
-75
-70
-65
-60
-55
-50
-45
-40
Frequency (GHz)
Pow
er (
dBm
)
Gaussian monocycle pulse shaped by a minimax FIR
filter.Filling efficiency: 74.96%
Gaussian monocycle pulse shaped by an elliptic
IIR filter. Filling efficiency: 68.29%
Both filters have 17 multipliers.
DSP Group, EE, Caltech, Pasadena CA
Comparison Elliptic shaper and minimax FIR shaper
0 2 4 6 8 10 12-55
-50
-45
-40
Frequency (GHz)
Pow
er (
dBm
)Elliptic IIR shaper has sharp transition band but cannot compensate the nonflatness of the transfer functions.
Minimax FIR shaper has the flexibility to compensate the nonflatness. But the transitio
n band is wide.
We can combine these two ideas to get both of their benefits.
DSP Group, EE, Caltech, Pasadena CA
IIR shaper design We divided the problem into two parts.
The first part is designing the Elliptic IIR filter H1
to fit the transition band of the mask .)( fM
The second part is designing the minimax FIR filter H2 to fix the nonflatness of the transfer functions . )()()(
2fSfHfG meq
).()()()()()()(22
22
2
22
1 fMfSfHfGeHeHfS meq
fT
nif
T
ni
DSP Group, EE, Caltech, Pasadena CA
0 2 4 6 8 10 12-60
-55
-50
-45
-40
Frequency (GHz)
Pow
er (
dBm
)
Results
Minimax FIR shaper: efficiency = 74.
96%
Elliptic IIR shaper: efficiency =
68.29%
Combination method: efficiency
= 78.92%
All shapers have 17 multipliers.
Combination method uses
7 multipliers on minimax FIR shaper and
10 multipliers on Elliptic IIR shaper.
DSP Group, EE, Caltech, Pasadena CA
Transient response
The impulse response of the FIR shapers has a duration of 2.4ns.
The proposed method has only 1.5% of energy outside this duration.
The transient response is small.
0 2 4 6 8 10-0.1
-0.05
0
0.05
0.1
0.15FIR
imp.
res
p.
0 2 4 6 8 10-0.2
-0.1
0
0.1
0.2New method
imp.
res
p.
time(ns)
1.5% of the energy
DSP Group, EE, Caltech, Pasadena CA
Conclusions
The pulse design is to generate a pulse such that radiated power can be maximized.
The IIR based pulse shaper is introduced. An elliptic IIR filter and a minimax FIR filter ar
e combined to fit the mask and the transfer functions.
The transient response of the proposed IIR filter is small enough to be neglected.
DSP Group, EE, Caltech, Pasadena CA
References
Terry P. Lewis, Robert A. Scholtz, “An ultrawideband signal design with power spectral density constraints,” Proc. 38th IEEE Asilomar Conf. on Signals, Systems, and Computers, pp. 1521-25, Nov. 2004.
X. Luo., L. Yang, and G.B. Giannakis, “Designing optimal pulse-shapers for ultra-wideband radios, ” Proc. of IEEE Conf. on Ultra Wideband Systems and Technologies, pp. 349-353, Nov. 2003.
B. Parr, B. Cho, K. Wallace, and Z. Ding, “A Novel Ultra-Wideband Pulse Design Algorithm,” IEEE Comm. Letters, pp. 219-221, 2003.
DSP Group, EE, Caltech, Pasadena CA
Thank you.