IEEE 802.14-95/094

TITLE: INTRA FM Modems
AUTHOR: Dr. W. J. Miller

ABSTRACT: Concepts, equations and calculated efficiencies are presented for frequency modulated CATV modems operating in non-Gaussian noise at very high bandwidth efficiencies. An entirely new network service for subscribers is also defined, based on transparent information transformations in AM and FM modems.

The following paper is based on Document # IEEE 802.14-95/094 prepared to assist and submitted to the IEEE 802.14 WG Cable TV Protocol Working Group September 12, 1995.

An FM modem exhibits a receiver gain, G, due to several effects that are unavailable to PM, AM or QAM modems. The S/N of the baseband signal out of an FM discriminator can be much larger than the C/N into the receiver. That is

S/N = G * C/N
where
G = Gm * Gd * Gr

Gm is the modulation gain	---- determined by the modulation index and by the companding function
Gd is the de-emphasis gain	---- determined by the pre-emphasis filter
Gr is the noise-reduction gain	--- determined by the de-companding

A novel FM data modem {P} that works with any multiple access system can be designed with the following features:

Robustness in Non-Gaussian Noise	(Wavelet FM)
High Bandwidth Efficiency	(FM-DSB and FM-SSB)
Non-Coherent IF	(No Carrier Recovery)
Low Cost	(No A/D converter)
Low Co-Interference	(FM capture effect)
Compatible with Analog	(Commuting Operators)

The Reverse Channel

The FM Modem is proposed because of the adverse noise environment {E} on the reverse (home to head-end) channel. The modem uses a Wavelet Synthesizer to transform the data into a pre-emphasized baseband signal which is then non-linearly compressed --the Mu-Law was used in simulations-- and then input to an FM modulator. The modulation index, m, and bits per symbol, B, are chosen to maximize the data throughput.

Of course the VCO, shown above for simplicity, can be replaced by more suitable methods for generating FM at passband. As will be discussed later, the FM transmitter can be an FM Single-Sideband transmitter (FM-SSB) or the more familiar constant-envelope FM-DSB, which is also ideal for wireless using Class C switching amplifiers. For SSB the FM signal is AM modulated by the inverse-log of the Hilbert transformed baseband signal {C}.

A Wavelet Synthesizer is a "Vector Filter". A group of B data bits comprising one symbol is partitioned into the coordinates of a vector. The DATA vector is input to a linear-phase FIR Vector Filter whose coefficients are square matrices multiplying the contents of vector shift registers. The unnormalized matrices transform the DATA vectors into new vectors (i.e. change their length and rotate their orientation) to form, after summation, a resultant SIGNAL vector. Each coordinate of the output SIGNAL vector is confined to a (slightly overlapping) frequency subband. Only the in-band coordinates are used.

At the receiver another vector filter, the Wavelet Analyzer, performs a reverse transformation to recover the original DATA vector. If necessary. the coefficient matrices in the receiver may be adaptively equalized to correct for path distortion, since the Analyzer is a fractional-rate FIR filter.

The Forward Channel

A Wavelet Synthesizer/Analyzer pair has no entropy loss in the passband, and since the transition to stopband can be made arbitrarily small (at the expense of delay, as required by Shannon), the pair form an optimal baseband modem. The polyphase matrices of a Wavelet pair are commuting operators.

A very interesting feature of optimal "commuting-operator modems" is that when co-located and placed back-to-back in the "unnatural" direction, a bandlimited analog signal will pass thru both linear modems. The analog signal is sampled and "received" as an analog vector by the 1st modem and "demodulated" by the convolutional rotations of the vector filter into a DATA-domain vector; and then the 2nd modem counter-rotates back to the original in-band analog vector.

As an example, any analog signal can be digitally encrypted between filters 1 and 2, (back-to-back wavelet modems), and then, converted back to analog without expanding its bandwidth. Remarkably, there is no digital compression algorithm. A remote 3rd and 4th back-to-back filter-pair digitally decrypt the signal. Any Guassian-like noise introduced in transmission appears to the "listener"....as it would in any analog connection ...as low-level background.

If modems 2 and 3 operate at a higher rate than modems 1 and 4, then the original analog signal can be error protected by, say, inserting a parity coordinate, thereby increasing the vector dimension (and transmitted bandwidth out of modem 2). This provides error-free analog transmission over a noisy link for ANY in-band analog signal (even the modem output of a FAX Machine).

Furthermore, a non-commuting FM (AM) passband modulator can be placed between filters 2 and 3 with no affect on the transparency (except pre/de-emphasis). In other words, modems 2 and 3 can be FM-Modems of the type described herein with vector filters 1 and 4 being additional filters used only when transmiting uncompressed analog. A spreading sequence can also be applied between filters 1 and 2 for DS-SS. Of course, compression algorithms and multiple-access arrangements are possible too.

This leads to the astonishing observation for IEEE 802.14 networks that uncompressed analog voice or video can be treated as DATA.

For instance using "cheap" (no DSP) set-top modems, Digital Forward Error Correction of uncompressed voiceband analog would work over CATV with existing telephones and FAX ---but with error-free digital quality. This contrasts with digital compression (i.e 13kbps "GSM" transmits only voice, at 23kbps with error correction, and suffers delay induced echoes). Aided by the good bandwidth efficiency, many new possibilites are opened to the system designer contemplating two-way voice, video and data networking over CATV.

FM Modem Design Details

For FM-DSB modems the modulation gain is the same as for textbook FM {C}

Gm = 6(m+1)(m^2)/(PAR)^2

where PAR is the peak-to-rms voltage ratio into the frequency modulator. Modem simulations using linear-phase wavelet filters {S} show the ratio squared is about 8 db before compression and 2 db after Mu-Law compression. In addition, the noise added in transmission undergoes a noise-reduction. This S/N improvement, Gr, in the receiver should be roughly equal to the compression of the rms in the transmitter. Noise reduction occurs when the inverse Mu-Law attenuation is applied at the receiver. This is illustrated below

In more detail, the Receiver (again neglecting the non-coherent IF stages) consists of a discriminator, inverse Mu-Law attenuator, and Wavelet Analyzer.

An FM discriminator has a noise probability density function that is proportional to the square of the frequency. The larger the deviation the more noise; therefore, the baseband Wavelet transformation places most of the data information at the lower baseband frequencies. This is called pre-emphasis.

If the baseband data signal is divided among M-1 coordinates of an M- schedule that matches the f^2 noise power density of the dimensional vector in such a way as to allow a pre-emphasis discriminator then

Gd = Gs * M^3/(M-1)

where Gs may be 1 in a carefully designed modem. With a little less care (a cheaper design?), the factor Gs varies slightly for each subband. The factor Gs(i) can be calculated by integrating f^2 over one subband; namely, from i to i+1. Designating H(i) as the pre-emphasis amplification in subband I

Gs(i) = H(i)^2/(3i^2+3i+1) for subbands i=0 to M-1

By a simple design procedure, subband i=0 carries k bits of data and no pre-emphasis so H(0)=1. Then subband 1 carries k-1 bits with H(1)=2 etc.. A typical assignment for M=8 that results in nearly equal power in each subband (a further requirement) would be

TABLE I
PRE-EMPHASIS (M=8)
i	0	1	2	3	4	5	6	7
bits	6	5	4	3	3	3	2	0	B=26
H(i)	1	2	4	8	8	8	16	1
TOTAL = 26 Bits per Symbol

Band 0 sends 6 bits using one of 64 levels, namely +- 1,3,5....
Band 6 uses only 4 levels, namely +-16 and +-48. That is , H(6)=16 is the pre-emphasis amplification at the transmitter for Band 6. The pre-emphasis and de-emphasis operations are really automatic...there are no separate filters.

For this particular example, each symbol transmits 26 data bits. The optimum symbol size depends on the C/N ratio. It is possible to simultaneously send an orthoganal baseband synchronization "clock bit" in band 7 for baseband timing recovery. An FM receiver does not need a carrier timing recovery loop. For low cost, the A/D converter may even be eliminated by counting zero-crossings in the IF stage (for moderate data-rates).

Since the modem uses k-bit multilevel signalling, the S/N ratio out of the Wavelet filter at the receiver must exceed the desired 2-level baseband ratio by a gain, Gk, to get the same BER as a 2-level baseband system.

Gk = (2^(2k) -1)/3

Then the required baseband S/N is

S/N = Gk * (2*Eb/No)

Where Eb/No determines the BER for 2-level baseband transmission.
Since S/N = G * C/N, FM modems require

Gk * (2*Eb/No) < Gm * Gd * Gr * C/N

Recalling that Gm depends on (m+1)m^2, this non-linear condition can be met by a specific choice of modulation index, m. Based on a preliminary design, a Table of the optimum m versus E is shown below, where E=(C/N)/(2Eb/No).

The Bandwidth Efficiency, R/W, is found from the Carson bandwidth, W=2(m+1)w. The partially unused subband makes w less than (M/2)*(R/B). R is the data rate and R/B is the symbol rate. The FM-DSB bandwidth efficiency, eDSB, is

eDSB = B/[M(m+1)]    Efficiency in bits/sec/hz

The design procedure is to specify E, then find m, and finally compute eDSB. The efficiency for FM single sideband, eSSB, is twice eDSB ....but it requires 3db more power since Gm is halved.

 

TABLE II
FM-SSB EFFICIENCY (M=8)
(C/N)/(2Eb/No)	Bits/Baud	Index	Bits/Sec/Hz
10log(E)	B	mSSB	eSSB	BENCHMARK {1}
5.3 db	26	0.807	3.596	8-PSK
7.0 db	26	0.687	3.852	16-QAM
13.2 db	33	0.675	4.925	64-QAM
19.3 db	33	0.370	6.022	256-QAM
25.0 db	40	0.382	7.233	1024-QAM
30.0 db	47	0.425	8.246	?
35.0 db	47	0.255	9.363	?
{2}

TABLE III
FM-DSB Efficiency
	M=4		M=8
10log(E)	B	eDSB	B	eDSB
25 db	18	3.564	40	3.899
30 db	21	4.062	47	4.476
35 db	21	4.478	47	4.958

 

Notes:

1. The theorectical limit for n-PSK and n-QAM efficiency is log2(n).
2. To reduce adjacent-channel interference a modem can dynamically reduce the modulation index without affecting the data rate or error rate, provided it is above its design operating point, E, in TABLE II.

Estimated Performance

Since Eb/No is set by the BER and, in practice, can be diminished by using forward error correction, it is appropriate to normalize to 2*Eb/No. Thus the results below will use the ratio, E, for the FM modem operating point

E = (C/N) / (2*Eb/No)    Operating Point

To compare the FM-modem to multilevel QAM at the same BER, one examines the Table at the value of E corresponding to E=Gn where Gn=(n-1)/3 for n-QAM. The corresponding traditional modulations are shown as benchmarks for comparison.

QPSK efficiency is typically 1.5 to 1.8. TABLE II shows single sideband efficiency could be considerably better than the theoretical limit for traditional modulations at low C/N but not as good at medium C/N. At very high C/N, FM-SSB could still be implemented while QAM would be overly complicated.

TABLE III examines FM-DSB at very high power for vector spaces of 4 and 8 dimensions. Very high C/N is considered on CATV since good forward-channel TV reception needs a C/N of over 45db. For the reverse channel, FM modems with long symbol times should offer excellent tolerance for non-Gaussian noise, so the FM operating point can be less conservative than for a traditional modulation. And there is no need for a large amplifier backoff allowance with constant envelope FM-DSB. In the Tables, an operating point of E=30 db is roughly a Guassian C/N of 45db at a BER of 10^-7 without FEC.

Everyone is familiar with the noise performance of FM versus AM radios in a thunderstorm. So on CATV reverse channels, FM modems could work very well.

 

Notes:

{C}_Couch; "Digital and Analog Communications Systems"; 4th ed Macmillan 1993

{E}_Eldering,Himayat,Gardner; "CATV Return Path Characterization for Reliable Communications"; IEEE COMMUNICATIONS Mag, Aug 1995

{P}_If included in a standard, the INformation TRAnsformation (INTRA) technology and INTRA modems described herein will be licensed on reasonable and non-discriminatory terms and conditions.

{S}_Soman,Vaidyanathan,Nguyen; "Linear Phase Paraunitary Filter Banks: Theory Factorizations and Designs"; Vol 41 No 12, IEEE Trans SP, Dec 1993.