NON-UNIFORM SIGNALING OVER BAND-LIMITED CHANNELS: A Multirate Signal Processing Approach. Omid Jahromi, ID:

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1 NON-UNIFORM SIGNALING OVER BAND-LIMITED CHANNELS: A Multirate Signal Processing Approach ECE 1520S DATA COMMUNICATIONS-I Final Exam Project By: Omid Jahromi, ID: Systems Control Group, Dept. of Electrical and Computer Engineering, University of Toronto ABSTRACT It is well known that if the communication channel is bandlimited to W Hz one can transmit at most 2W independent symbols per second. Generalized sampling theorems, nevertheless, do not restrict such a signaling to be uniform [1], [2], [3]. That is, it is theoretically possible to distribute the data symbols non-uniformly in time as long as the average signaling rate does not exceed 2W. Since the average baud rate remains the same, non-uniform signaling techniques are rarely used in normal communication scenarios. Recently, based on the observation that almost all of the public telephone network is digital, it has been proposed that 56 kb/s transmission is theoretically possible over voice band channels [7]. The proposed signaling is a nonuniform one which makes it possible to receive independent data symbols using available 8 khz PCM samplers while the average baud rate is kept limited to 7 kbaud (W=3500 Hz). In this report, inspired by the results given in [4], a general frame work for non-uniform signaling over band limited channels is proposed. The analysis tool here is multirate signal processing theory using which it will be shown that when the channel is sampled with a rate higher than the Nyquist rate, M out of L output samples are allowed to be linearly independent as long as MF L W S / 2. It is interesting to note that any subset of M samples in a frame of length L will work (which seems to be in contrast with the results claimed in [8].) 1

2 I. Introduction Fig. 1 shows a simple block diagram representation of a simplex modem operating over a voice band channel. As depicted in the figure, it is generally assumed that the channel is an ideal band limited one with bandwidth khz and the noise is additive and Gaussian (SNR about 30 db). Fig. 1 A simple model for voice band data transmission. Assuming the nominal bandwidth and SNR values given above, one can easily show that the channel capacity would be in the range of 30 kb/s [6]. However, the actual voice band channels are almost entirely digital. (Only the short subscriber access links are analog.) So, a more realistic channel model can be considered as in Fig. 2. Fig. 2 A more realistic channel model showing internal digital transmission. The slicer in the above figure represents the 8 bits A/D converter which, because of companding, has nonuniform step sizes. If we could send 8k independent pulses per second and if the pulses were designed such that at each sampling time they exactly fell within one slicer level, then we could send data at rate 64 kb/s! However, this is not possible for two reasons: 2

3 1. The band limited nature of the channel (W= 3500 Hz) allows transmission of up to 7k symbols per second only. 2. The slicer levels and the sampling times are not known at the transmitter. The later issue is a serious practical challenge. To the author s best knowledge, even using training sequences at the initial phases of operation (as proposed in [7]), there is no established method available to estimate the sampling instants and slicer levels [10]. Nevertheless, in order to proceed further, we assume that such information are somehow available to the transmitter!! The first problem (i.e. suitable signaling under the channel bandwidth constraint) is not trivial either. Assume that the PCM modulator in Fig. 2 generates exactly the same amplitude levels assigned to the slicer levels. Also, assume that the PCM modulator is proceeded by a sufficient order interpolator to smooth out the output pulse shapes. Now, if we design the pulses generated by TX so that they fall within the correct thresholds at the slicer input, we can be hopeful that the same pulses will be regenerated at the PCM modulator output (within some minor errors due to interpolation). The channel bandwidth limitation, however, does not allow the TX to transmit a signal such that all of the samples taken at the rate 8kHz (at the slicer input) have appropriate amplitude. In other words, since the channel is oversampled, the amplitude of the samples taken at the slicer would be linearly dependent to the amplitude of past and present samples too. Note that even decreasing the signaling rate to the Nyquist rate (7 kbaud) wont help us control the amplitude of the samples taken at 8kHz. So, as a solution to the above conflict, we have to use non-uniform signaling at 8 kbuad where information bearing samples are transmitted at a local rate higher than the Nyquist rate but mixed with some dummy samples appropriately chosen to keep the bandwidth limited. In the following sections, using multirate signal processing techniques, we will describe a general method to embed/retrieve independent samples in/from a larger set of (linearly dependent) samples for transmission over a bandlimited channel. We will see that the average signaling rate can not exceed the Nyquist rate anyway. The report is organized as follows: In section-i the method proposed in [7] is outlined. Section II describes some generalized sampling theorems in discrete-time domain. In section III the results of section II are used to develop a general discrete-time framework for non-uniform signaling over band limited channels. In section IV a continuous-time PAM/Multipulse equivalent is presented. Section V explores time domain properties of the pulses associated with the equivalent PAM/Multipulse modulator. 3

4 Section VI describes some generalizations and extensions to the case of multiband bandpsss channels. Finally, conclusions are made in section VI. NOTE: The notation used in this report are somewhat different from those adapted in [4] and [8]. II. Uniform and Nonuniform Sampling in Discrete Time Consider a continues time signal x(t) whose Fourier transform is bandlimited to W Hz. We can exactly reconstruct x(t) from its samples x(nt) using the familiar interpolation formula n= π x( t) = x( nt) sinc( ( t nt)) T n= (1) where 1 2W. Assume that 1 L = 2 T T M W, L > M, then we are able to re-sample the discrete time signal x(n)=x(nt) further while we are still able to reconstruct the original continuous-time signal exactly. In this connection, the following facts/theorems are well known [5], [9]: Fact-1: If we uniformly sample the bandlimited continuous time signal x(n) with 1 L 2 T = M W, L > M, then the resulting discrete time signal x(n) would be bandlimited toω p M = π. L Fig. 3 (a) Sampling rate conversion in discrete time (b) The required ideal low pass filter. Fact-2: If the discrete-time signal x(n) is band limited to ω p M = π, then one can re-sample x(n) further L by a factor M L without causing aliasing distortion (i.e. without loss of information). A procedure for such sampling rate conversion is shown in Fig. 3. The corresponding frequency domain interpretations are given in Fig. 4. As depicted in Fig. 4, the ideal lowpass filter between the upsampler and the decimator ensures elimination of image spectra that occur because of upsampling. This is required to prevent aliasing that occurs when the up sampled signal is decimated again [9]. Therefore, 4

5 since aliasing is prevented, we can use the interpolation formula (1) to recover the original continues time signal directly from the decimated signal. This is achieved simply by assuming a new sampling period T ' L = M T. Fig. 4 Frequency domain representation of sampling rate conversion operations (a) Fourier transform of the original discrete-time signal (b) Fourier transform of the up-sampled signal (c) after low-pass filtering (d) after decimation. Combined together, facts 1 and 2 lid to: Theorem-1: One can always represent the signal at the output of a bandlimited channel using 2W uniformly-spaced samples per second. Increasing the sampling rate to say 2W L M (L > M) will produce a discrete time signal which is band limited to M L π. On the other hand, we can interpret x(t) as the signal outgoing from a band-limited channel and x(n) a discrete time symbol sequence which is to be transmitted through it. The above theorem could then be re-stated as: 5

6 Fig. 5 Periodic non-uniform sampling demonstration. (a) the original signal (b), (c) and (d) examples of nonuiformly decimated signals obtained from (a) by keeping only two samples out of each block of 5. Theorem-2: It is possible to transmit 2W L M uniformly spaced samples per second (L > M) through a bandlimited channel if and only if the discrete symbol sequence is bandlimited to M L π. Now, we turn our attention to nonuniform sampling of discrete time signals. The following theorem is the key result in this case [4]: Theorem-3: A discrete-time signal x(n) bandlimited to M L π, can be perfectly reconstructed using periodic non-uniform sampling, i.e. by retaining an arbitrary set of M samples out of blocks of length L. Fig. 5 provides a simple example demonstrating non-uniform sampling in discrete time. In the following, we describe a method for nonuniform sampling and reconstruction. This method also serves as a proofby-construction for the above theorem. The derivations closely follow those provided in [4]. Consider the structure shown in Fig 6(a). For an input block of length L, the decimator output in each branch generates only one sample. So, the M branches in Fig 6(a) totally generate M samples per each block. By proper choice of the delay coefficients n 0, n 1,..., n M 1, one can obtain all possible M-sample subsets out of blocks of length L. This structure, therefore, represents a general nonuniform sampler in discrete time. 6

7 Fig. 6. Discrete time structures for nonuniform sampling and reconstruction. In [4], inspired by the results available for achieving perfect reconstruction in multirate M channel filter banks, it is proposed to use the structure of Fig 6(b) to perfectly reconstruct the original signal x(n). Now, the question is whether one can specify a set of interpolator filters F,..., F n0 n M 1 so that x(n)=x(n)? Hopefully, the answer is in the affirmative. The rest of this section outlines the procedure using which we can specify the interpolating filters. Using standard multirate signal processing techniques (See for example [5] pp. 224) we can obtain the transfer function from x(n) to x(n). It turns out that M 1 n k X( z) = A ( z) X( zw ) k= 0 k (2) where W = e j2π/ M and A k ( z) are given by A k 1 L 1 n j kn j ( z) = z W Fn ( z), 0 k L -1 (3) j L j= 0 Note that 0 n 0 n 1... n M-1. In order for X(z) to be equal to X(z), (2) implies that A k ( z) = 0, 1 k L - 1 A ( z) = 1, k = 0 k (4) Combining (3) and (4) we get M 1 kn n j j W z F ( z) = Lδ ( k), 0 k L -1 j= o n j (5) 7

8 The above constraints represent a system of L linear equations in M unknowns which, in general, does not have any solution (since M < L). However, not all of the equations should be satisfied if X( e jω ) is M M band limited. In fact, if we divide the frequency range π ω π into M-1 consecutive regions, L L each of length 2π L and numbered from 0 to M-1, it turns out that only M equations (instead of L) need to be satisfied at any region [4]. After some algebra, we end up with the following system of M equations in M unknowns for the ith region: W W... W W W... W n0 n1 nm 1 ( M 1) n 0 ( M 1) n1 ( M 1) nl 1 0 i n... 0 ( zw ) F ( z) n 0 0 i n1 ( zw ) Fn1 ( z) L... = i n 0 M 1 ( zw ) Fn ( z) M (6) where the L on the right-hand side occurs at the ith row. The M by M matrix on the left-hand side is a Vandermonde matrix with distinct columns and is hence nonsingular. This establishes the existence of a unique set of synthesis filters at every region. We must solve the above system of equations for all frequency regions (i.e. for all 0 i M 1) to determine the set of interpolating filters F n0,..., F n M -1 completely (i.e. for the whole frequency range). Since it is always possible, at least in theory, to specify the set of M interpolating filters satisfying (6), the above arguments serve as a proof for Theorem-5 as well. One can show that the interpolating filters obtained by solving (6) have piecewise constant frequency responses and satisfy the conjugate symmetry property as well [4]. Therefore, they can be represented using real coefficients in time domain. III-Non-uniform signaling using multirate signal processing techniques Sampling a band-limited signal and sending discrete samples (symbols) through a band-limited channel are essentially the same. So, we can use the non-uniform sampling structures introduced in the previous section for non-uniform signaling over band-limited channels too. As mentioned previously, there is little motivation for doing so in normal situations. However, such techniques might be valuable in special cases like the one addressed in Section-I. 8

9 The proposed structure for nonuiform signaling through an oversampled band-limited channel is depicted in Fig. 7. The signal samples of x(n) are assumed to arrive every T x second. They are converted into blocks of length M by the set of M decimators and, after that, up-sampled by a factor L. It is easy to show that arrival of a new sample of x(n) provides one non-zero input (followed by L-1 zeros) for one of the interpolating filters. In other words, only one of the filters is exited by arrival of any new sample in x(n). The filters, nevertheless, have infinite impulse response and generate long output sequences at the new sampling rate 1 L 1 =. These sequences are finally added up to generate the symbol sequence y(n). T M T y x Fig. 7 A multirate DSP structure for non-uniform signaling over band-limited channels. The symbol sequence y(n) is band-limited to M L π as described in the previous section. So, interpolating its samples using sinc pulses with repetition rate 1 1 = will result in a continues time signal T T y S bandlimited to M L T S 1. Therefore, no ISI will occur as long as M 1 L T s 2W (7) Note that our physical signaling rate 1 T S can be up to L times faster than the Nyquist rate (2W) while M the actual information signaling rate ( 1 T x ) always remains less than or equal to it. 9

10 A DESIGN EXAMPLE: Consider an ideal band-pass channel band limited to 3kHz (say, the telephone channel). We can transmit 6k samples per second by transmitting one sample every 1/6 ms using ordinary discrete-to-continuous conversion (interpolation using sinc pulses given in (1) with T= 1/6 ms). We can also use uonuniform signaling. For example, we can send 8k symbols per second while 6 symbols in each block of 8 represent information symbols and the two others are determined by the nonuniform modulator itself to keep the continuous time signal band limited to 3kHz. In this example, we design a nonuiform modulator of the type shown in Fig. 7 to do the job. For the present case we have M=6, L=8, 1 T S = 8 khz, 1 T x = 6 khz. We can choose the information bearing samples to be any arbitrary subset of 6 samples from blocks of length 8. However, an elegant choice here is to divide each block into two smaller blocks of length four. Then, we can choose three samples in each smaller block to be those bearing information (i.e. we let M=3 and L=4). This is advantageous because we only need three digital filters to construct the non-uniform modulator of Fig 7 now. We also assume that the first three samples in the blocks (now of length four) are to carry information. (One can arbitrarily assume other cases as well). This assumption implies that n 0 = 0, n = 1 and n = Using these parameters, (6) enables us to specify the three digital filters F ( z), F (z) and F n2 ( z). Note n0 n1 that (6) should be solved three times to specify the filter characteristics for the whole frequency range. The resulting filters are: 3 π 3 π 1 + j π < ω π < ω 4 4 jω π ω jω Fn ( e ) 2 ω π -j π = <, e Fn ( e ) = 0 < ω π, π 3 π 3 1 j < ω π 2 < ω π π 1 j π < ω 4 4-2jω jω π e Fn ( e ) = 2 < ω π π 3 1+ j < ω π 4 4 (8) Note that the filters have piecewise-constant characteristics in frequency domain and satisfy the conjugate symmetry property. So, they can be implemented in time domain using real coefficients. Having the three filters calculated, our design is almost complete. Using these filters in the transmitter of Fig. 7, we can send the information signal x(n) nonuniformly interpolated in time through the channel. 10

11 The interesting point is that the receiver is very simple and consists only of a non-uniform sampling scheme! In other words, to retrieve the information bearing symbols, it is only required to sample the channel out put at the sampling points we had originally placed the symbols. This is what the receiver part in Fig. 7 exactly does. (See also Fig 8.) Fig. 8 PAM-Multipulse modulator equivalent to the multirate structure of Fig. 7. IV- Interpretation as PAM/Multipulse modulation The discrete-time non-uniform signaling scheme introduced in the previous section has a very close connection with multipulse modulation concepts in continuous-time domain. In this section we derive the equivalent continuous-time Multipulse/PAM modulator and show that the generalized Nyquist criterion is satisfied by this equivalent system. For the sake of simplicity, we use the system designed as an example in the previous section. The concepts, however, can be easily extended to the general case. To begin with, we notice that the digital filters F ( z), F (z) and F n2 ( z) can be combined with the sinc n0 n1 pulse generator p(t), to get a new set of analog pulse shapes we call g 0 ( t), g 1 ( t) and g 2 ( t). These pulses are easily characterized in frequency domain since g 0 ( t), g 1 ( t) and g 2 ( t) are just impulse responses of the analog (i.e. interpolated) versions of the digital filters F ( z), F (z) and F n2 ( z). That is to say, the n0 n1 Fourier transforms of g 0 ( t), g 1 ( t) and g 2 ( t) are exactly similar to the frequency responses of F ( z), F (z) and F n2 ( z) with the digital frequency range [0 π ] mapped to the analog frequency range n0 n1 [0 L M W Hz]. So, using (8), we can plot the Fourier transforms of g 0 ( t), g 1 ( t) and g 2 ( t) as in Fig

12 Fig. 9 Frequency domain characteristics of the analog pulsesg 0 ( t), g ( t) and g ( t) Note that in Fig. 9 it is assumed that T this figure, it is very easy to show that and S = M L so that the maximum frequency is M 1 = W. From 2W L T s G i ( jω) + G i ( j( ω W)) + G i ( j( ω + W)) = 8, for i = 0,1,2 (9) 3 3 * 4 * 4 4 * 4 G i ( jω) G j ( jω) + G i ( j( ω W)) G j ( j( ω W)) + G i ( j( ω + W)) G j ( j( ω + W) = 0, for i j (10) or, combined together 12

13 m= 2π * 2π G i ( j( ω m )) G j ( j( ω m )) = 8 δ i j (11) MT MT x Equation (11) shows that the pulses g 0 ( t), g 1 ( t) and g 2 ( t) satisfy the Generalized Nyquist Criterion (see, ex., [6] pp. 325) when they are excited by the samples of x(n) every M T x seconds. Based on the above observations, we can draw the equivalent PAM/Multipulse modulator as in Fig. 8. It is interesting that although each pulse is excited every M T x seconds, the pulses are not excited simultaneously. That is, each pulse is excited T x seconds before/after the next/previous pulse. x V- Time Domain Considerations As depicted in Fig. 9, the pulses g 0 ( t), g 1 ( t) and g 2 ( t) have piecewise constant Fourier transforms. So, they can be represented in time domain using modulated sinc pulses. In fact, since the frequency regions over which the pulses have constant characteristics are all of the same width, we can represent each pulse using one sinc function modulated by several sinusoidal functions: sin( 2π ( W 3) t) g 0 ( t) = A ( 1 sin( 2π( 2W 3) t) + cos( 2π( 2W 3) t)) 2π ( W 3) t sin( 2π ( W 3) t) g1 ( t TS ) = B (sin( 2π( 2W 3) t) + cos( 2π( 2W 3) t)) 2π ( W 3) t sin( 2π ( W 3) t) g 2 ( t 2TS ) = C ( 1+ sin( 2π( 2W 3) t) cos( 2π( 2W 3) t) 2π ( W 3) t (12) After some straightforward trigonometric manipulations and combining the multiplicative constants together, we end up with the following compact representation: g ( t) = C k k sin( 2π ( W 3)( t kts )) 2 sin( 2π ( W 3)( t kts )) (13) 2π ( W 3) t i= 0 i k M 3 where TS = =, C k is a scale factor and appropriate time-shifts are included to make the pulses 2WL 8W look like causal! The pulse shapes derived above are exactly the same as those derived in [1] or proposed in [8]. This is not surprising since the interpolating functions using which one can reconstruct a continuous-time signal based on its periodic non-uniform samples must be unique [1]. 13

14 The pulses given in (13) have interesting time and frequency domain characteristics. First, they satisfy the Generalized Nyquist Criterion. Seconed, it is very easy to verify that, in time domain, each pulse has unit amplitude at its excitation instant and is zero at the excitation instants of other pulses. These additional zeroes are provided by the multiplicative sinusoidal terms in (13) and are necessary for avoiding ISI at the non-uniform sampler (receiver) of Fig 8. A very important characteristic of the pulses derived for non-uniform signaling is that they could have large peaks where they are not forced to be zero [1]. This point should be considered very carefully since even a minor timing error at the receiver can cause a dramatic error in the amplitude of the received samples. In frequency domain, each pulse spans the whole available bandwidth (See Fig 9). So, the underlying PAM/Multipulse modulation can be thought of as non-uniform time division multiplexing. Fig. 10 An example of lowpass to multiband bandpass transformation. VI- Application to Multiband Bandpass Channels and Other Extensions The results presented in the previous sections assume a bandlimited ideal lowpass channel. However, they can also be extended to the case of a general multiband channel whose total bandwidth support is W Hz. This is because any lowpass signal can be transformed to a multiband bandpass signal as shown in Fig 10. So, we can transform our signal at the transmitter to fit the channel characteristics and then recover the original lowpass signal at the receiver using the inverse operation. Note that such a transformation does not affect the information transmission capabilities of the channel or the signals involved. We derived the pulse shapes in (13) based on the digital filter characteristics determined for the specific example described in section-iii. One can follow the same procedure to derive pulse shapes for any other situation from the digital filters obtained by solving (6). The fact that the resulting pulses serve as 14

15 functions interpolating samples taken in a non-uniform periodic manner, however, implies that the pulses would be always the same as the interpolating functions derived by Yen (See [1], eq. 9). VII- Conclusion We approached the nonuniform signaling problem from a discrete time point of view. This approach is appealing in view of the fact that both efficient techniques and rigorous supporting theories for manipulating multirate discrete time signals have been developed [5], [9]. These techniques include the so called polyphase structures and perfect reconstruction (PR) multirate filter banks. Using filter bank structures, the authors in [4] had proposed a technique for reconstruction of a band limited discrete time signal from its nonuniformly re-sampled version (see Fig. 7). In this report, inspired by nonuniform sampling techniques described in [4], we proposed a nonuniform signaling scheme in discrete time (Fig. 7). The discrete time modulator of Fig. 7, takes M data samples from x(n) and embed them in frames of L transmission samples constituting y(n) (L > M). The remaining L-M samples are determined by the digital filters F ( z), F (z) and F n2 ( z). Proper choice of n0 n1 these filters (as induced by equations (6)) enable us to transmit the composite sequence y(n) at a physical rate 2W L M which is higher than the Nyquist rate. As can be seen from (6) the choice of filters depends on the position of data bearing samples within a frame (That is, to the choice of delay coefficients n 0, n 1,..., n M 1.) These coefficients can be selected arbitrarily. In other words, we can transmit data samples arbitrarily positioned within the frames. This conclusion, which is in contrast with the points claimed in [8], results from the fact that the matrix at the left hand side of (6) is always invertiable. In spite of many efforts, the author could not find any compromise between these (seemingly) contradictory results. We then showed that the discrete-time nonuniform modulator of Fig. 7 is equivalent to the continuoustime PAM/Multipulse modulator of Fig. 8. The equivalent pulse shapes are derived and shown to satisfy the generalized Nyquist criterion. We also showed that the pulse shapes obtained by converting the solutions of equation (6) to continuous time belong to the general class of interpolating functions derived in [1] (See [1] eq. 9). For the sake of simplicity and clarity, the modulator structures are derived for the design example provided in section-iii. However, the procedures are quite general and the results are extendible to other cases as well. The nonuniform signaling techniques are of theoretical interest as generalizations of popular uniform ones. Practical interest in such methods is limited as they do not provide better transmission performance 15

16 over their uniform counterparts. Nevertheless, they might be of potential interest in some special scenarios like the one described in [7] for fast transmission over existing voice band channels. The author, however, believes that the scheme proposed in [7] is very hard to realize in practice because of many difficulties in synchronization and ADC level detection. In addition, we must have in mind that amplitude sensitivity of nonuniform interpolating pulses to timing errors is much higher than those of normal sinc pulses [1]. References [1] Yen, J. L., On nonuniform sampling of bandlimited signals, IRE Trans. Circuit Theory, Vol. CT-3, pp , Dec [2] Jerry, A. J., The Shannon sampling theorem- its various extensions and applications, Proc. IEEE, Vol. 65, No. 11, November 77, pp [3] Papoulis, A., Probability, random variables and stochastic processes, 3rd ed., McGraw-Hill, 1991 [4] Vaidyanathan, P.P. and V. C. Liu, Classical sampling theorems in the context of multirate and polyphase digital filter bank structures, IEEE Trans. Signal Proces., Vol. 36, No. 9, Sep [5] Vaidyanathan, P. P., Multirate systems and filter banks, Prentice Hall, 1993 [6] Lee, E. A, and D. G. Messerschmitt, Digital communication, 2nd ed., Kluwer Academic Publishers, 1994 [7] Kalet, I. et. al., The capacity of PCM voiceband channels, ICC 93, pp [8] Ayanoglu, E., Data transmission when the sampling frequency exceeds the Nyquist rate, IEEE Communications Letters, Vol 1, No. 6, pp , Nov [9] Crochiere, R. E. and L. R. Rabiner, Multirate processing of digital signals, in Lim and Oppenheim Eds., Advanced Topics in Signal Processing, Prentice-Hall, 1988 [10] Meyr, H., M. Moeneclaey and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing, Wiley Interscience,