New Formulas for Irregular Sampling of Two-Bands Signals255

we have a model for (observed) jitter quantified at the

value 1l. Of course, we can complicate the sampling

plan by introducing sampling gaps in the .

1

n

t

3.4. Example 4

Examples above deal with two samplings t1 and t2 with

equal mean rate 1. Following the value of (the place

of subbands) we can imagine samplings with mean rates

which are different and not multiple (but rational be-

tween them). For instance consider the following case

l

12

2

=, =,3,2.

3

nn

n

tntalla

This corresponds to

12

12

1

1

2

2

4

=0,=2,=2 ,=3

,=sincπ

3π2

,=sinc

23

nn

n

n

knllakn

gtt tn

n

gtt ta

l

2

2,n

tt is the usual sampling formula matched to the

sampling rate 3/2 delayed by , true for power spectra

in

a

3π2,3π2.

The larger the better the choice

for available samplings. Unlike the preceeding examples,

we are in a situation of a true oversampling (

l

is arbi-

trarily small). However, if is not too small, the mean

rate sampling is more favourable than the Nyquist one.

l

3.5. Example 5

One or both sequences can be mixed. For instance

12

for even

=,=,2,2

3for odd

4

nn

nn

ttnal

nn

.la

The formula (6) can be used when ,

with

2,2lla

11 2

12 212

1

1

2

2

3

=0,=4,=21,=2,=2

2

2for even

πcosπ6

π2π

,=sincos 3

23 for odd

πcos π8

,=sincπ

nn n

n

n

knlk lnlakn

n

tn n

tt

gtt

n

tn n

gtt tna

l

0

4. Conclusions

Most of the time, processes used in communications occ-

py symmetrical power spectral bands in the form u

=, ,,>ba aba

. Very often, the relative ban-

dwidth 2bab is small. However, most of the sam-

pling formulae are matched to baseband processes where

=,aa . In this case the choice of errorless samplings

is large, whatever the sampling, uniform or irregular [5,6,

13]. The sampling mean rate for errorless reconstruction is

πa in the latter case (the Nyquist rate) and it is

πba in the former case (the Landau rate) [1]. In

communications the Landau rate is small in front of the

Nyquist rate. The research for errorless samplings with

Landau rate is important for reducing calculus cost. The

choice of errorless samplings is limited to the PNS [9,10]

and has to be increased. It is the aim of this short paper. A

new sampling formula is proved and examples are given.

They are based on formulas true in baseband and generally

well-known [4,14,15]. Example 3 deals with irregular

samplings at the Landau rate and can be used in the pres-

ence of jitter. In example 4, we have two samplings with

different periods which generalizes the PNS2. The method

which is used can be generalized to other power spectra

including more than two pieces [16,17]. It is also possible

to use a mixing of several periodic samplings for the se-

quences t1 and/or t2 [11,12].

REFERENCES

[1] H. J. Landau, “Sampling, Data Transmission, and the

Nyquist Rate,” Proceedings of the IEEE, Vol. 55, No. 10,

1967, pp. 1701-1706. doi:10.1109/PROC.1967.5962

[2] B. Lacaze, “About Bi-Periodic Samplings,” Sampling Th-

eory in Signal and Image Processing, Vol. 8, No. 3, 2009,

pp. 287-306.

[3] B. Lacaze, “Equivalent Circuits for the PNS2 Sampling

Scheme,” IEEE Circuits and Systems, Vol. 57, No. 11,

2010, pp. 2904-2914. doi:10.1109/TCSI.2010.2050228

[4] A. J. Jerri, “The Shannon Sampling Theorem. Its Various

Extensions and Applications. A Tutorial Review,” Pro-

ceedings of the IEEE, Vol. 65, No. 11, 1977, pp. 1565-

1596. doi:10.1109/PROC.1977.10771

[5] B. Lacaze, “Reconstruction Formula for Irregular Sam-

pling,” Sampling Theory in Signal and Image Processing,

Vol. 4, No. 1, 2005, pp. 33-43.

[6] B. Lacaze, “The Ghost Sampling Sequence Method,”

Sampling Theory in Signal and Image Processing, Vol. 8,

No. 1, 2009, pp. 13-21.

[7] H. Cramér and M. R. Leadbetter, “Stationary and Related

Stochastic Processes,” Wiley, New York, 1966.

[8] A. Papoulis, “Signal Analysis,” McGraw Hill, New York,

1977.

[9] J. R. Higgins, “A Sampling Theorem for Irregular Sample

Points,” IEEE Transactions on Information Theory, Vol.

22, No. 5, 1976, pp. 621-622.

doi:10.1109/TIT.1976.1055596

[10] J. R. Higgins, “Some Gap Sampling Series for Multiband

Signals,” Signal Processing, Vol. 12, No. 3, 1987, pp. 313-

319. doi:10.1016/0165-1684(87)90100-9

Copyright © 2011 SciRes. JSIP