Communications Theory and Engineering Master's Degree in Electronic Engineering Sapienza University of Rome A.A. 2018-2019
TDMA, FDMA, CDMA (cont d) and the Capacity of multi-user channels
Code Division Multiple Access (CDMA)
CDMA and Spread Spectrum CDMA is based on a technique called spreadspectrum As its name indicates this technique consists in spreading the spectrum over the whole set of available frequencies All users transmit then over all frequencies but are separated from one another thanks to coding
Direct Sequence Spread Spectrum C x x CODING y Bandwidth of the input signal frequency Bandwidth of the coded signal frequency
Direct Sequence Spread Spectrum T b bit time Original signal (band related to bit rate=1/t b ) Codeword: 0110001001 called spreading sequence Coded signal (band related to chip rate=1/t c ) T c chip time
Direct Sequence Spread Spectrum Signal 1 Coded signal 1 Signal 2 Coded signal 2 Adding coded signals 1 and 2
Direct Sequence Spread Spectrum Spreading sequence used to encode signal 1 Received signal X Signal 1 multiplier Decoded signal It works if codes are orthogonal!
Direct Sequence-CDMA: reference scheme Transmitter S Digital signal CDMA coder (multiplier) CDMA signal Digital to analog: pulse shaping, modulation S TX Transmitted signal Code generator
Coding in DS-CDMA binary data signal Codeword DS-CDMA-coded signal s ( j) t ( ) ( j) k ( t) ( ) c j [ ] s DSCDMA Generated bit stream for each user Assigned Codeword Binary signal after coding for each user 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 1 0.5 0-0.5-1 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 5 10 x 10-3 0 2 4 6 8 0 0.005 0.01
Effect of coding Generated bit stream for each user Assigned Codeword Binary signal after coding for each user 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 1 0.5 0-0.5-1 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 5 10 x 10-3 0 2 4 6 8 0 0.005 0.01 If f b is the bit rate of the source data stream then the bit rate of the coded stream is G. f b where G is the coding gain The band of the coded stream is G times the band of the source signal
The DS-CDMA coded signal ( j) s DSCDMA s ( j) t ( t) j = a k Digital binary signal k ( ) = a k j k N DS : length of the codeword T C : chip time ( ) N DS m=1 ( ) δ t kt ( ) DS-CDMA-coded signal c ( j) [ m]δ ( t mt C kt ) Spreading Signal
Example Source: PCM coded digital speech Bit rate of source: 64000 bits/s Minimum band is: 32 khz DS-CDMA coding with codeword length 100 The coding gain G is 100 The bit rate of the DS-CDMA PCM stream is 64. 10 kbits/s Minimum band is: 3.2 MHz
DS-CDMA: reference scheme Receiver S RX Received signal Front-End filter and demodulator Code generator Multiplier Integrator or adder to the decision unit The receiver performs a correlation It is called correlation receiver
The correlation receiver The correlation receiver implements the correlation operation Given two discrete-time real sequences x[k] and y[k] The auto-correlation functions of x[k] and y[k] are: R xx [k]= + x[m]x[m k] and R yy [k]= y[m]y[m k] m= The cross-correlation functions are: R xy [k]= + m= R xy [k]= R yx [ k]= x[m]y[m k] + m= + m= y[m]x[m k]
CDMA : the partial correlation problem Partial correlations prevent the receiver to totally cancel the contributions of other users even in the presence of spreading codes having low cross-correlation In presence of partial correlations, the received signal is therefore affected by Multi User Interference The partial correlations can be reduced by proper choice of the spreading codes, but cannot be totally eliminated CDMA system capacity is thus tipically limited by Multi User Interference, rather than by thermal noise.
What is Multi User Interference (MUI)? RX Device #4 wireless transmission Device #2 RX Device #1 TX TX Device #3
What is Multi User Interference (MUI)? MUI is generated by the presence of several users sharing a same resource Ideally, if multiple access was well-defined this interference would not exist since all users would be orthogonal in the resource space The presence of MUI depends on the robustness of the multiple access scheme to phenomena that cause loss of orthogonality between users
CDMA : the near-far problem If all users transmit at the same power level, then the received power is higher for transmitters closer to the receiving antenna Thus, a transmitter that is far from the intended receiver may be strongly at risk due to interference from other users that are close to that receiver This problem can be mitigated by introducing power control by which transmitters adjust their transmission power so that power arriving at a receiving antenna is equal for all transmitters In other words, the nearby transmitters are assigned with a lower transmit power level than the far away transmitters Power control can be easily achieved in centralized access schemes (e.g. cellular networks), and is a challenging issue in distributed systems
MUI in TDMA-based networks TDMA is usually adopted in centralized network organizations In these networks one can reasonably suppose that MUI can be neglected by proper design of the guard times between time slots guard time packet TS j TS j+1 TS j+2 time
MUI in FDMA-based networks If not well-designed FDMA suffers from inter-channel interference between adjacent channels that is a form of MUI Thus the need for guard bands and consequently loss in efficiency of use of the frequency resource With frequency guard bands one can suppose as in TDMA that MUI is negligible Note that here users do not need to be coordinated and that this scheme applies to distributed topology of access and distributed network organization
MUI in CDMA-based networks The downlink in a centralized network In the case of an access point transmitting to N u mobile receivers, signals may be encoded using orthogonal signature codes The N u signals are perfectly synchronized at TX, so that basically they arrive synchronous at each mobile receiver Each receiver can demodulate its own signal with negligible interference from the other signals sharing the same bandwidth.
MUI for uplink CDMA The uplink in a centralized network Case of an access point receiving from N u mobile nodes, that use orthogonal signature codes. The N u signals may be perfectly synchronized at TX, as in synchronous networks, and perfectly orthogonal thanks to a good design of the code space These signals may arrive out of phase as in TDMA: this effect can be adjusted by the RX, based on an exchange with the transmitters during which the RX asks (as in TDMA) to adjust clock phases Different signals experience, however, different channel conditions and this provokes a loss of orthogonality that cannot be easily recovered The above effect is the main reason for MUI in CDMA networks and is present regardless of network organization
System model for MUI analysis Data sequence a 1 [n] Encoder & Transmitter Transmitter 1 code 1 Encoded signal s 1 (t) Received useful signal s RX1 (t) h 1 (t) + P TX1 P RX1 Thermal noise n(t) r(t) Receiver Transmitter 2 code 2 s 2 (t) P TX2 h 2 (t) s RX2 (t) P RX2 + s i (t) MUI signal Transmitter K code K s K (t) P TXK h K (t) s RXK (t) P RXK
MUI estimation under the SGA Decision variable Z = Z u + Z mui + Z n Cumulative noise term System performance can be easily evaluated under the Standard Gaussian Approximation (SGA) hypothesis: the cumulative noise term (Z mui + Z n ) is treated as an additive white Gaussian noise term Average BER at receiver output under the SGA 1 BER = erfc 2 γ ( γ ) SNR tot depends on the modulation format SNR tot = E ( ) 2 b = Z u 2 σ n 2 σmui σ 2 n Eb + σ Variance of Z n Variance of Z mui 2 mui
Capacity of Multiple Access Techniques A reminder: what is channel capacity according to Shannon. Channel capacity C in bits/s for a band-limited Additive White Gaussian Noise (AWGN) ideal channel with a band-limited and average power-limited input is given by: & $ 1 + % P C = W log2 WN 0 #! " P: average power W : bandwidth of the input signal WN 0 : noise power N 0 : unilateral thermal noise density power
Capacity of Multiple Access Techniques Note that P is an average power and therefore: P = CE b where E b is the energy per bit under the condition that the transmission rate matches the channel capacity. Given the expression of channel capacity, one can easily find: E b N 0 = C/ W 2 1 C/W C/ W 2 C/W
Capacity of Multiple Access Techniques : FDMA Suppose N u FDMA users. Each user is allocated with a bandwidth W/N u and transmits power P n =P/N u. Therefore capacity C n for user n is: C n W & P # & = $ + n W N P log 1! = log $ 1 + u n 2 Nu % 0 ( )! $! 2 W Nu N 0 " Nu % W N " # System capacity C for the network of N u users is: & N # & = = $ + upn P N! = $ ucn W log2 1 W log 1 + % W N 0 " % W N 0 C 2 #! " which shows that total capacity is equivalent to the case of a single user using power P = N u P n and all bandwidth W. Note that when N u increases, C increases. Bandwidth allocation for a single user becomes however smaller.
Capacity of Multiple Access Techniques : TDMA In TDMA, each user is allocated with a Time Slot of normalized duration 1/N u. Each user transmits within its allocated time over the overall bandwidth W using total power P. The capacity per user is therefore the same as in FDMA: 1 & P # W & P = W log $ +! = $ 2 1 log 1 + Nu % W N 0 " Nu % W N 0 Cn 2 #! " When compared to FDMA: note that each user transmits with power P although for a shorter time P = N u P n When N u increases, there is a practical limit for P beyond which a single user cannot reasonably operate.
Capacity of Multiple Access Techniques : CDMA In CDMA, each user transmits over the total bandwidth W with power P n. Let us consider two cases: Case A - Users are non-cooperative (they ignore each other) Case B - Users are cooperative (they know each other and coordinate with one another)
Non-cooperative CDMA In this case at each receiver the signal originating from the (N u -1) non-useful users are perceived as interfering noise. The capacity per user is thus: C n = W log 2 & $ 1 % + W N 0 P + n # ( N )! u 1 Pn " The total capacity is: C = N u C n = N u W log 2 & $ 1 % + W N 0 P/N + u # ( N )! u 1 P/Nu " Note that the relation of C to N u is more complex than in FDMA and TDMA.
Cooperative CDMA In the case of cooperative users we can suppose that all users are synchronized. The receiver knows all codes of all users and can jointly detect all signals with no interference between users. The total channel capacity is therefore: & N # & = $ + upn P W log! = $ 2 1 W log 1 + % W N 0 " % W N 0 C 2 which is the same of TDMA and FDMA. #! " However, there is a fundamental difference in the present case when compared to TDMA and FDMA
Cooperative CDMA The capacity of the single user is not in this case equal to a fraction C/N u of the total capacity, rather, it is equal to: C n & $ 1 + % P N = u W log2 W N 0 #! " which can be shown to be greater than in the TDMA / FDMA case & P N # & = $ + u 1 P W log! > $ 2 1 W log 1 + % W N 0 " Nu % W N 0 Cn 2 #! "
Cooperative CDMA The aggregate rate R is thus bound by C: N u R = R i < C = W log2 & $ 1 + % P W N i= 1 0 #! " while the rates of the single users must satisfy: R i & $ 1 + % P N < u W log2 W N 0 #! "
Example: N=2 Comparison of capacity regions