IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 12, DECEMBER 1996 1629

Bandwidth Efficient PAM Signaling Using Wavelets Jay N. Livingston and Chien-Cheng Tung

Abstruct- In this article, we demonstrate a bandwidth effi- cient method of pulse amplitude modulation (PAM) signaling using a family of orthonormal wavelets as the baseband pulse. These wavelets can be transmitted using single side-band (SSB) transmission, since they have zero average value. We provide a comparison with raised-cosine signaling, with the wavelet ap- proach offering 50% greater data rate at the same bandwidth.

I. INTRODUCTION NE of the most important families of wavelets to come 0 out of wavelet research is that of orthonormal wavelets

[ 11-[4]. These wavelets have the important property that shifts of the wavelets are orthogonal to the parent wavelet, as are dyadic expansions of the parent wavelet.

The property of orthonormality appeals to the communica- tions engineer, since in signal design we are concerned with representing multidimensional signal constellations using or- thogonal functions. In this work, we propose a simple signaling scheme using wavelets as the orthonormal waveforms used to represent the signal constellation. In particular, we are able to construct a system that is more spectrally efficient than raised-cosine signaling.

The remainder of this work is outlined as follows. In Section 11, we provide some basics on orthonormal wavelets and their use in communications system. In Section 111, we describe the proposed system and give its performance parameters. Section IV provides a comparison to raised cosine signaling, and Section V concludes the paper.

11. WAVELET FUNDAMENTALS

The orthornomal wavelet is a family of functions, each related by either a translation in time, or a scaling in time, or both. We write the wavelet as

(1) hmn(t) = 2-"12 . h(2-"t - n )

where m , n E 2. Here, the parameter 2-" is performing a time scaling of the wavelet h( t ) . The parameter n is performing a time shift of the wavelet. The most important property of these wavelets, for our purposes, is the property of orthonormality over translations and dyadic expansions. We may write this property compactly as

(hmn(t) , hkZ(t)) = &nkSnZ . (2)

Throughout this paper, we will make use of what we will call a composite signal. This signal will consist of the parent

Paper approved by F A Marvasti, the Editor for Data Communications of the IEEE Communications Society. Manuscript received July 14, 1994; revised October 27, 1995 This work was supported in part by NSF Grant

The authors are with the Department of Electrical Engineering, Texas A&M

Publisher Item Identifier S 0090-6778(96)090 19-8.

NCR-9314221

University, College Station, TX 77843-3 128 USA

wavelet, and its translates, as well as a given number of dyadic expansions and their translates.

Let us assume that we wish to transmit the sequence a,, n = 0 ,1 , . . . , where a, E {fl}. We can transmit this data using the parent wavelet as the shaping pulse used in the modulation. We can also transmit a second sequence, b,, n = 0,1, . . . , where b, E {zt l} , by using the first dyadic expansion of the parent wavelet as the shaping pulse modulated by the data. Adding the two signals together yields

n m

It is of most importance to note that (ho,, him) = 0, i.e., the two signals y(t) and ~ ( t ) are orthogonal.

We may continue this process, adding dyadic expansions to transmit N data streams, to yield

N

m=O n N

= amn2TmI2 . h(2Tmt - n) (4) m=O n

where amn is the mth data sequence to be transmitted using the mth dyadic expansion of the parent wavelet.

To recover the data after transmission, all that needs to be done is to project the signal z ( t ) onto the family of wavelets used to generate the composite signal, generating the transmitted sequence plus channel noise. This is the same process used in any matched filter receiver. The major advantage again being the computationally efficient pyramid decomposition available for our use. This method also allows us to perform matched filtering by using only two filters in an iterative fashion (see [4]).

111. SYSTEM DESCRIPTION A block diagram of the proposed communications system is

given in Fig. 1. Note that we are proposing a single side-band (SSB) transmission scheme for digital data.

Of particular importance is the spectral efficiency of the scheme. The wavelet we will use is found in [4] and is given in Fig. 2(a). This particular wavelet was chosen because of its narrow bandwidth compared to most other wavelets in the literature. The digital representation of this wavelet is orthog- onal to itself over shifts by multiples of four samples. Hence, the minimum symbol period T corresponds to four samples of the wavelet. Given this fact, the normalized spectrum of the parent wavelet is given in Fig. 2(b). Note that the sampling rate is four times the symbol rate. The measure of bandwidth for our purposes will be when the signal is 70 dB down from

0090-6778/96$05.00 0 1996 IEEE

1630 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 12, DECEMBER 1996

Cos( wt )

Sin( wt )

Fig. 1. SSB communications system used for the example.

0.6

-0.6 -40 .35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40

Sample Number

(a)

Normalized Frequency

(b)

Fig. 2. Parent wavelet and its power spectral density (PSD) (a) 22 tap wavelet, sampled four times/symbol period and (b) power spectrum of 22 tap wavelet.

its maximum value. This yields a bandwidth for the wavelet of 0.355, normalized to the sampling frequency, or 1.42lT Hz, referred to the symbol rate.

For comparison, we will use a raised cosine pulse with a p = 0.2, transmitting data at the same symbol rate. The spectrum of this pulse is given in Fig. 3. Note that its bandwidth is 0.15, referred to the sampling rate, or 0.6lT Hz, referred to the symbol rate.

Taken on a bandwidth basis, the wavelet system compares unfavorably. However, we will now exploit the fact that the

-0.254 I . I . I 7 7 I I I I , -35 -30 -25 -M -15 -10 -5 0 5 10 15 20 25 30

Sample Number

(a)

-40

5

-120 -I 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 (

Normalized Frequency

(b)

50

Fig. 3. Raised cosine pulse and its PSD: (a) raised cosine pulse sampled four timedsymbol period (beta = 0.2) and (b) power spectral density for raised cosine pulse.

wavelet has zero average value and use SSB transmission. Using SSB, the wavelet scheme, after modulation, has a band- width of 1.42lT. The raised cosine pulse, having significant frequency content at 0 Hz, will need to be transmitted using double side-band. Its bandwidth will then be 1.2lT.

At first glance, it appears that the raised cosine system will still outperform the wavelet system. However, by using the first dyadic expansion of the parent wavelet, we may transmit an additional 112 blsymbol period in the same bandwidth. In terms of spectral efficiency, the wavelet system can transmit

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 12, DECEMBER 1996 1631

System

Raised Cosine

Wavelet 1

Spectral Efficiency (low-pass) Spectral Efficiency (band-pass)

0.83 b/S/Hz n.a. d; = d f - 2 I Z ; ~ ( 5 ) 0.70 b/S/Hz 0.86 b/S/Hz i f 0

Wavelet 1.5

Wavelet 1.75

1.5 b/l.42/T Hz, versus 1 b/1.2/T Hz by the raised cosine system. Note that the energy per bit remains the same (each wavelet is an orthonormal function). In fact, we may continue to add dyadic expansions, reaching (in the limit) an efficiency of 2 b/l.42/T Hz, while retaining the same energy per bit. If we try to transmit 2 b per symbol using the raised cosine pulse and PAM signaling, we increase the energy per bit by a factor of 2.5. In Table I, we list a comparison of the spectral efficiency of the wavelet scheme versus the raised cosine scheme. It can be seen that relatively few dyadic expansions are required to achieve superior efficiency.

We note in passing that the above bandwidths were com- puted based on the signals being low-pass in nature. Since they are actually bandpass, the bandwidths, referred to the sampling rate, are really 0.29, 0.325, and 0.34, for the parent wavelet, the parent wavelet plus the first dyadic expansion, and the parent wavelet plus the first two dyadic expansions. Thus, the wavelet system can be considered to be even more efficient than we have presented it. This is illustrated in Table I, where we have included the spectral efficiency for the case where the wavelet is considered as a band-pass signal.

1.06 b/S/Hz 1.15 b/S/Hz

1.23 b/S/Hz 1.29 b/S/Hz

IV. PERFORMANCE OF WAVELET SSB

The system described above was simulated digitally to observe the effects on the wavelet due to the SSB modulation. The wavelets were generated using the techniques described in Section II. To observe the effects more carefully, oversampling by a factor of four was performed @e., the symbol period corresponded to 16 samples).

A discrete time Hilbert transform was used to generate the SSB signal. The length of the filter was 201 taps. The output corresponding to two different low-pass filter designs in the demodulator was examined. The effect of filtering is to introduce intersymbol interference (IS) into the received pulses. We may compute a worst-case effect of the IS1 by considering the effects of transmitting a sequence that causes maximum interference with receiving a specific pulse. The worst-case sequence is one that causes each received IS1

where d f is the Euclidean distance between points in the original constellation, zi are the IS1 coefficients, and d; is the worst-case minimum distance between points taking into account the ISI.

A. 0.1 dB Ripple Design

The first design has a pass-band ripple of 0.1 dB, and a filter length of 59. IS1 is present, but the terms are small, amounting to 0.12% of the total energy. For this example, d; = 1.84, amounting to a loss in performance of 0.73 dB.

B. 0.01 dB Ripple Design The second design has a bandpass ripple of 0.01 dB, and

a filter length of 85 taps. For this example, d; = 1.95, amounting to a loss in performance of 0.21 dB.

In conclusion, we note that an equalizer could be used to compensate for some of the ISI, leading to improved performance.

V. CHANNEL CODING

Instead of using a dyadic expansion to transmit an extra 1/2 b per symbol period, we can use the extra capacity to support a channel code. For example, if we transmit using the parent wavelet and the first dyadic expansion, we may use a rate 2/3 channel code. This yields a system that transmits one information bit per symbol period. A simple four-state code can then deliver 3 dB of coding gain over the raised-cosine system, with essentially no bandwidth penalty.

VI. CONCLUSION

In this work, we have presented a communications system that uses wavelets as the orthonormal basis for signaling. We have shown that for a specific design, namely a SSB sys- tem, wavelets compare favorably to other bandwidth efficient schemes.

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[2] G. Strang, “Wavelets and dilation equations: A brief introduction,” SIAM Review. vol. 31, no. 4, DD. 614-627, 1989.

keying (BPSK) signal constellation to have moved closer to design,” IEEE Trans. Signal Processing, vol. 40, no. 9, Sept. 1992.