Weighted PNS sequences for digital alias-free processing signals

Weighted PNS Sequences for Digital Alias-Free Processing Signals

DONGDONG QU, ANDRZEJ TARCZYNSKI

Department of Electronic Systems

University of Westminster

115 New Cavendish Street

London, W1W 6UW, UK

Abstract: - In this paper Weighted Periodic Nonuniform Sampling (WPNS) for Digital Alias-free Signal

Processing is proposed. The work is a direct extension of previous research on Periodic Nonuniform Sampling.

First, the methodology of measuring the level of aliasing within the required range of frequencies is proposed.

Then the optimal WPNS is found by searching a carefully selected subspace of feasible solutions and applying

Lawson algorithm for weight calculation. It is shown that WPNS has better alias-suppression properties than

traditional PNS. Both real- and complex-valued weights are considered.

Key-Words: - Digital alias-free signal processing, PNS, Nonuniform sampling, Lawson algorithm, Aliasing

1 Introduction Digital Alias-free Signal Processing (DASP) is a

novel DSP approach that aims at use of nonuniform

sampling to suppress the effect of aliasing. DASP

provides means of processing signals in wide

frequency ranges that are not limited by the half

sampling rate. Therefore, comparing with classical

DSP, DASP normally uses lower sampling rates [1,2].

This feature makes DASP a suitable tool for solving

signal processing problems at wide frequency ranges

for which maintaining high sampling rate is not

viable either from technical or economic point of

view. One of the earliest ideas of DASP has been

reported in [1]. In that paper Shapiro and Silverman

have shown that by use of random sampling it was

possible to estimate the Power Spectral Density (PSD)

of stationary signals. This effect was achievable even

if the bandwidth of the signal exceeded arbitrarily

much half of the average sampling frequency.

Thorough reviews of various DASP techniques can

be found in [2]-[4]. Reported applications of DASP

include spectral analysis [5]-[7] and software radio

[3]. As mentioned before, DASP requires that the

analysed signals are sampled nonuniformly.

However, not each nonuniform sampling scheme is

suitable for DASP. The processing algorithms used

by DASP are typically more complex than those used

in classical DSP. Therefore, DASP is mainly

recommended for high frequency applications [8]

where traditional approaches cannot be deployed due

to excessive demands on the high sampling rates.

To explain how DASP is related to more

traditional DSP approaches, we show Figure 1. The

figure compares the average sampling rate (α ) for

collecting data with some benchmark frequencies

related to the processed signals. Depending on the

position of α , different signal processing

methodologies should be chosen. The benchmark

frequencies are: max2 f , SSFB2 and B2 . Here, maxf

denotes the lowest frequency that is known to be

above all frequencies present in the processed signal.

SSFB is the total length of positive frequency

intervals that are known to contain the whole signal

spectrum and, finally, B is the actual single-sided

bandwidth of the signal spectrum. For example, if the

signal consists of two bandpass components, the first

placed somewhere between 90 and 110MHz and the

other between 120 and 150MHz. Then,

150max =f MHz and 50=SSFB MHz. To calculate

B , one should know the actual bandwidth of the

components. E.g. if the actual position of the first

component is ]100,95[ MHz and the actual position

of the second component is ]135,130[ MHz, then B

is 10 MHz. When α exceeds max2 f , the signal can

be processed using uniform sampling and classical

DSP. If α ∈ ]2,2[ maxfBSSF , then uniform sampling

can be used only in some special cases. A universal

solution is to combine Periodic Nonuniform

Sampling (PNS) with specialized polyphase signal

processing. Further reduction of the sampling rates

brings more challenge to DSP. When α < SSFB2 ,

uniform sampling cannot be used at all. Special

sampling techniques that utilize PNS and/or random

sampling combined with sophisticated processing

algorithms must be deployed. It can be shown that as

long as α > B2 it is still possible to reconstruct the

processed signal and hence perform on it usual signal

Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp1-6)

processing tasks [9]. When α < B2 , only selected

signal processing tasks that do not require full

reconstructability of the signal can be accomplished.

In this paper, we extend our previous results [10]

on use of periodic sampling schemes in DASP. The

extension consists in application of weights in the

PNS sampling function. We investigate the effect of

using real- and complex-valued weights. We

demonstrate that use of weights in PNS sequences

can further reduce the level of aliasing in the

processed signal. We also revisit the problem of

measuring the residual aliasing in (W)PNS. The

approach proposed here allows incorporation of

bandpass spectral support of the processed signal in

PNS/WPNS design and can be easily extended to

arbitrary multiband support. We use Weighted Least

Square (WLS) [11, 12] and Lawson algorithm to

design the optimal WPNS sequence.

In the next section we introduce some properties

of the discrete signal sampled with use of WPNS. In

section 3, method of measuring the level of aliasing

in the sampled signal is proposed. Section 4 describes

the way of optimally designing WPNS sequence.

Finally, two numerical examples are presented to

illustrate the methodology advocated in this paper.

2 Weighted Periodic Nonuniform

Sampling (WPNS) The concept of PNS has been introduced in [10,

13]. Here we extend it to a “weighted” version. In this

section we define WPNS and present some analyses

that demonstrate how this sampling method leads to

suppression of aliasing. Consider a periodic sequence

of sampling instants defined by nnmN tmTt +=+ ,

Nn ,,1L= , ,+∈ ZN Zm∈ , where T is the

period of sampling, N is the number of samples

within each period, and nt are the sampling instants

inside the first period. Let )(tx be a continuous time

signal to be sampled. The discrete-time signal

obtained by Weighted Periodic Nonuniform

Sampling of )(tx is defined by:

)(txd = ∑∞

−∞=−

n nn ttwtx )()( δ (1)

where InRnn jwww += is a periodic sequence, with

period N , of complex-valued weights. Note that

when traditional PNS is used, 1=nw . We refer to

∑∞

−∞=−=

n nn ttwts )()( δ (2)

as the sampling function. Figure 2 shows sample

plots of )(ts for PNS and WPNS. Since )(ts is

periodic, it can be expanded into Fourier series

∑∞

−∞==

k

tT

kj

kectsπ2

)( . Hence its spectrum is given

by

∑∞

−∞=

−=k k

T

kfcfS δ)( (3)

where .1

1

2

∑=

−=

N

n

tT

kj

nk

n

ewT

cπ

The spectrum )( fX d

of the discrete-time signal (1) is the convolution of

spectrum )( fX of )(tx , and )( fS . Hence

∑∞

−∞=

−=k kd

T

kfXcfX )( (4)

In a perfect case we would have 10 =c and 0=kc if

0≠k . This would simplify (4) to )()( fXfX d = .

Unfortunately, designing such sampling sequences,

that are completely free of aliasing, is impossible.

Therefore, in this paper we confine ourselves to

identifying “unwanted” coefficient kc and

minimizing them, rather than zeroing their values.

Shifted images 0, ≠

− kT

kfXck of the original

signal spectrum )( fX can be considered as residual

aliasing. In the next section we discuss how to

measure the level of such aliasing.

3 Measuring the Level of Aliasing Let [ ] [ ]maxminminmax ,, ffff ∪−− be a

conservative estimate of the spectral support of )(tx .

Therefore, 0)( =fX if minff < or maxff > . It

follows from this observation that

−T

kfX 0= if

TffkTff )()( minmin +<<− , Tffk )( max−<

or .)( max Tffk +> By denoting

,)()( max1 Tfffk −= ,)()( min2 Tfffk −=

Tfffk )()( min3 += , and Tfffk )()( max4 += ,

we get

∑

∑

=

=

−

+

−=

)(

)(

)(

)(

4

3

2

1

)(

fk

fkk k

fk

fkk kd

T

kfXc

T

kfXcfX

(5)

Our goal in designing WPNS is to make )( fX d a

good approximation of )( fX . Taking into account

Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp1-6)

that )( fX is a Hermitian symmetric function that

zeros outside [ ] [ ]maxminminmax ,, ffff ∪−− it

suffices that the approximation holds only inside

[ ]maxmin , ff . Then )( fX can be estimated in

[ ]minmax , ff −− by ),()( * fXfX d −≈ and for

frequencies outside the spectral support by

.0)( =fX By varying frequency f inside

[ ]maxmin , ff , we note that the appropriate )(1 fk

changes from ( ) Tff maxmin − to 0, )(2 fk from 0

to ( ) Tff minmax − , )(3 fk from Tfmin2 to

( ) Tff maxmin + and )(4 fk from

( ) Tff maxmin + to Tfmax2 . Therefore, in order

to provide a good approximation of )( fX by

)( fX d inside [ ]maxmin , ff , we need to minimize all

sck ' whose index Gk∈ , where the set G is

defined as the following:

{ [ ]( ) ( ) [ ] }0,,

2,2:

minmaxmaxmin

maxmin

≠−−∪

∈=

kTffTff

TfTfkkG

.

We propose the following measure of the level of

aliasing:

=∈

0

maxc

cJ k

Gk (6)

Since ∑=

=N

n

nwT

c1

0

1, we request that Tw

N

n

n =∑=1

then 10 =c and our criterion simplifies to

{ }kGkcJ

∈= max (7)

The measure (7) will be used to design the optimal

WPNS sampling sequence.

4 Optimum WPNS Sequences When designing an optimal PNS/WPNS

sequence we need to take into account a number of

practical constraints that limit the space of feasible

solutions. First, for practical reason we assume that

all sampling instants are multiples of some short time

interval L . The second constraint is motivated by the

fact that Analogue to Digital Converters (ADC) need

some minimum time between two consecutive

samples. Therefore we request that the distance

between sampling instants is not shorter than H .

We find the optimal sampling sequence by

solving the following optimization problem. Given:

L , rLH = , pLT = , minf and maxf , where r

and p are integers. Determine the number of the

sampling instants N inside one period T , the

positions of those instants:

Lmt nn = , Nn ,,1L= (8)

and weights nw so that cost (7) is minimized and the

following constraints are satisfied:

rmm nn +≥+1 for 1,,1 −= Nn L (9)

prmmN ≤+− 1 . (10)

Note that (8) guarantees that each sampling instant is

a multiple of L while (9) and (10) impose the

required minimum distance between sampling

instants. The weights nw could be either complex- or

real-valued.

The algorithm for solving the above problem is

stimulated by the design of PNS sequences in [10].

We note that the set of feasible solutions is the same

as in the previous paper and that the same small set of

feasible solutions needs to be searched in order to

find the optimal one. The difference between [10]

and the current approach is that now in order to assess

the quality of a sampling sequence we need to find

the appropriate weights nw for which (7) is

minimized. The algorithm implementing this step is

described below. Let T

Nww ],...,[ 1=w and

Tt

T

kjt

T

kj N

ee ],...,[22 1 ππ −−

=kz then k

TzwT

ck1

= .

Since we request that TwN

n

n =∑=1

, we get:

T=+= I

T

R

TTwjewewe (11)

where ]1,...,1[=Te . Direct minimization of index (7)

with respect to weights nw is not a straightforward

task. Therefore we propose to use an iterative

procedure that follows the principles of the Lawson

algorithm [14]. Instead of directly minimizing (7) we

solve a sequence of Weighted Least Square

problems:

*TMww

2

2 1

TcJ

Gk

kkWLS ==∑∈

λ (12)

where ∑∈

=Gk

k

H

kkzzM λ . The coefficients kλ are

iteratively changed as recommended by the Lawson

algorithm, )()()1( ncnn kkk λλ =+ . After a few

iterations the weights nw that minimize (12) are

almost identical with those minimizing (7). In order

Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp1-6)

to find the minimum of (12) while satisfying

constraint (11) we form the following:

I

T

R

T*T

IR

weweMww

ww

212

21

)(1

),,,(

λλ

λλ

+−+

=

TT

L

(13)

By calculating the derivatives of (13) with respect to

Rw and Iw and equating them to zero, we get

01

)(1

),,,(

122

21

=++

=∂

∂

eMwMw

w

ww

*TT

R

IR

λ

λλ

TT

L

(14)

01

)(1

),,,(

222

21

=+−

=∂

∂

eMwMw

w

ww

*TT

I

IR

λ

λλ

jTT

L

(15)

It follows from (14) and (15) that

=

=

−

05.0

5.0

5.0

5.0

00

00

2

2

1

2

2

2

1

2T

T

T

T

T

0

0

w

w

Aw

w

e0

0e

e0MM

0eMM

I

R

I

R

T

T

RI

IR

λλ

λλ

(16)

where RM and IM are real and imaginary parts of

M . To calculate Rw and Iw from (16) we partition

matrix A into subblocks. Hence,

=A

=

−

0D

CB

e0

0e

e0MM

0eMM

T

T

RI

IR

00

00. Then we

calculate the inverse of this matrix as:

−

−=

−

−−−−−

GGDB

CGBCGDBBBA

1

1111

1 (17)

where ( ) 1-1CDBG

−= . Then

=

−

0

TCGB

w

w1

I

R. (18)

Note that if only real-valued weights are required

then similar analysis yields:

( )( ) eMMe

eMMw

1TT

1T

−

−

+

+=T . (19)

5 Numerical Examples In this section, we present two numerical

examples that illustrate the concept of WPNS. In both

examples, we assume that the conservatively

estimated spectral support of the analysed signals is

40 – 145MHz, hence 105=SSFB MHz. The signals

are sampled using Analogue to Digital Converter

MAX1190 by MAXIM / DALLAS [15]. Although

the bandwidth of this converter is 400MHz its

maximum sampling rate is only =sf 120MSps,

which is at least 1.75 times too slow to process the

signal within the frequency range of the spectral

support. In both experiments, we use the following

parameters: the minimum distance between sampling

instants nsfH s 33.8/1 == and

nsHL 67.15/ == .

In the first experiment we compare cost (7) for

PNS and WPNS with real- and complex-valued

weights. We vary the period T between

nsT 65.41min = and nsT 125max = in steps of L .

The results are presented in Figure 3. The dotted,

dashed and solid lines show the quality of WPNS

with complex-valued weights, WPNS with

real-valued weights and PNS. We notice that WPNS

provides better performance than PNS.

In the second experiment we demonstrate how

the signal spectrum can be reconstructed from

subsampled data when WPNS and DASP are used.

Our test signal in this experiment consists of two

bandpass components. The actual position of the first

component is [ ]100,95 MHz, and the actual position

of the second is [ ]80,75 MHz. Hence, the total

bandwidth of the signal is 10=B MHz. The test

signal is:

)10150sinc(150)10160sinc(160

)10190sinc(190)10200sinc(200)(

66

66

tt

tttx

×−×+

×−×=.

The experiment is carried out as if this information

was not available and only conservative estimate of

spectral support 40 – 145MHz is known. First we

design the optimal WPNS sequence for nsT 125= .

Then we use the sequence to collect 120 samples of

the signal inside a window of length sT µ25.110 = .

The average sampling rate in this case is

96=α MHz. Next the spectrum of the discrete time

signal

∑=

−=120

1

2)()(

n

ftj

nndnewtxfX

π (20)

is calculated for frequencies between 40 – 145 MHz.

Its magnitude is shown in Figure 4. As we can see

from the plot, there are two clearly identifiable

bandpass components. Using these results we obtain

more accurate approximation of the spectral support

(though still conservative):

Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp1-6)

∪∉

∪∈=

MHz]105,90[]85,70[,0

MHz]105,90[]85,70[,1)(

f

ffF . Note

that the bandwidth of the new approximation of the

spectral support is 30MHz, much less than half of the

sampling rate .α Therefore now we can effectively

use other specialized algorithms such as Minimum

Energy Reconstruction (MER) [16] to process the

signal. Figure 5 shows the magnitude of the spectrum

of MER that is constructed from the collected data

and the spectral support )( fF . The ripples on the

signal spectrum are the result of signal windowing

and are hardly caused by the smeared aliasing. The

comparison of )( fX T the spectrum of the

continuous-time signal truncated to the observation

window and the estimated spectrum of MER

)( fX D in the range of frequencies 70 – 110MHz

shows that both spectra are almost identical. In fact

the RMS error is less than 0.02%.

6 Conclusions In this paper we propose use of Weighted

Periodic Nonuniform Sampling (WPNS) for DASP

applications. The method is a direct extension of [10]

where PNS without weights has been used.

Introduction of weights has improved the level of

suppressing aliases in the sampled signal.

The proposed approach to signal processing

consists of two stages. In the first stage a rough

approximation of the signal spectrum is constructed

and used to significantly refine the spectral support of

the analysed signal. Since at this stage we often deal

with small signal to “smeared aliasing” ratios (see e.g.

Fig. 4), any improvement in alias suppression like

this offered by WPNS over PNS is welcome. Once

the spectral support of the signal is estimated more

accurately the signal can be processed (reconstructed,

Fourier transformed, filtered, etc) using classical

approaches for nonuniformly sampled signals.

Fig. 1: Average sampling rate α and its associated

sampling schemes

0 20 40 60 800

0.5

1

s(t)

t

0 20 40 60 800

0.5

1

s(t)

t

(a)

(b)

Fig. 2: One period of s(t): (a) PNS and (b) WPNS with

real-valued weights

41.67 83.33 1250.35

0.45

0.55

0.65

0.75

0.8

T [ns]

J

Fig. 3: Measure of (7) against length of period T, given

5/HL = : complex-valued weights case (dotted line),

real-valued weights case (dashed line) and no weights

case (solid line).

max2 fB2 SSFB2

Uniform Sampling

PNS or Uniform Sampling

Random Sampling or PNS

Random Sampling

0α

DASP DSP

[Hz]

Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp1-6)

40 60 80 100 120 140-145

-140

-135

-130

-125

-120

-115

-110

Frequency [MHz]

|X(f)| [dB]

Fig. 4: )( fX d estimated on the optimal WPNS

sequence.

40 60 80 100 120 140-160

-150

-140

-130

-120

-110

-100

Frequency [MHz]

|X(f)| [dB]

Fig. 5: The reconstructed signal spectrum.

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http://para.maxim-ic.com/compare.asp?Fam=F

ast_ADC&Tree=ADConverters&HP=ADCDA

CRef.cfm&ln=

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Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp1-6)