S. Verd, "On the channel capacity per unit cost," IEEE Transactions on Information Theory, vol. 36, pp. 1361--1391, September 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....transmitted signal satis es lim ### #(# ; ## )## =0. The peak constraint results in Appendix B and the Gaussian input results imply that for low SNR, Rayleigh fading channels are at a capacity disadvantage as compared to Rician fading channels for equal values of #. But, it has been shown in [2, 18] for single antenna transmit and receive channel Rayleigh fading provides as much capacity as a Gaussian channel for low SNR. We next extend that result to multiple transmit and receiveantenna channel for the general case of Rician fading. The result for Rayleigh fading will follow as a special ....

....fading will follow as a special case. ####### # Let # beRician (3.1) and the receiver have no knowledge of #. For xed # , # and # Proof: First, absorb into # and rewrite the channel as with the average power constraint on the signal ##[tr# # ##= ## . It has been shown [18] that if the input alphabet includes the value 0 (symbol with 0 power) for a channel with output # , and conditional probability denoted by #(# ##) #( #(# = #) 0) is the set of values that the input can take, ## is the average power constraint on the input (in our case, ....

S. Verdu, \On channel capacity per unit cost," IEEE Trans. on Inform. Theory, vol. 36, no. 5, pp. 1019-1030, September 1990.

....[26] specialized Gallager s results to the Rayleigh fading channel and obtained the capacity divided by energy as a function of bandwidth and signal energy, concluding from this that the in nite bandwidth Rayleigh fading channel has the same capacity as the in nite bandwidth AWGN channel. Verd u [27] considered capacity per unit cost for general cost functions and derived a simple expression for the capacity per unit cost for memoryless channels for certain cost functions. Kennedy [15] considered the capacity per unit time of diffuse WSSUS fading channels. Using an M ary frequencyshift ....

....a Vector Rayleigh Channel In this section the theory of capacity per unit cost is applied to derive a basic information inequality for a vector Rayleigh fading channel. A. Background: Capacity Per Unit Cost We brie y review the notion of capacity per unit cost in this section, following Verd u [27]. Consider a discretetime channel without feedback and with arbitrary input and output alphabets denoted by A and B respectively. An (N; M; code is one in which the blocklength is equal to N ; the number of codewords is equal to M ; each codeword (x m1 ; xmN ) m = 1; M , ....

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S. Verdu, \On channel capacity per unit cost," IEEE Trans. Info. Th., vol. IT-34, pp. 1019-1030, Sept. 1990.

....and showed the similar conclusion that the mutual information goes to zero as the bandwidth gets large. Subsequent to the conference version of this work [11] Hajek and Subramanian [5] have obtained more recent results by applying the theory of capacity and reliability function per unit cost [3] [13] to related problems. By using a certain fourthegy of the signal as a cost measure (related to the fourth moment) they showed that the mutual information achieved by spread spectrum signals in fading channels is small because their fourthegy is small. For a comprehensive survey of other results ....

S. Verd, "On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. 36, pp. 1019--1030, Sept. 1990.

....82 Corollary VI.2. For purely rayleigh fading channels Gaussian input gives lim ae 0 C=ae = 0. The peak constraint results and the Gaussian input results give an indication that for low SNR rayleigh fading channels are at a disadvantage compared to rician fading channels. But, it has been shown in [40, 67] for single antenna transmit and receive channel rayleigh fading provides as much capacity as a Gaussian channel with the same energy for low SNR. We will extend that result to multiple transmit and receive antenna channel for the general case of rician fading. The result for rayleigh fading will ....

....r max (HmH y m ) N(1 Gamma r) Proof: First, we rewrite the channel as X = H S W with the average power constraint on the signal S to be given by E[trf S S y g] ae M TM = aeT . In other words, we have absorbed p ae M into S. It has been effectively shown ([67]) that if the input alphabet (the set of values the input signal S, can take) includes the value 0 (symbol with 0 power) for a channel with output X, with condition probability given by p(XjS) then lim PC 0 C P C = sup s2S D( p(XjS = s) k p(XjS = 0) P s where S is the set of values that ....

S. Verd'u, "On channel capacity per unit cost," IEEE Trans. on Inform. Theory, vol. 36, no. 5, pp. 1019--1030, September 1990. 148

....This reduces the multiple antenna channel with T 1 to a single antenna Rayleigh fading channel with T = 1 and N times the received SNR per antenna. As is well known, the low SNR capacity of such a channel is SNR N log 2 e, achieving the above upper bound. See for instance Example 3 in [9]. Thus, lim SNR 0 CM;N (SNR) SNR = N log 2 e (bps=Hz) The above analysis shows that the non coherent and the coherent capacities are asymptotically equal at low SNR. Hence in the low SNR regime, there is no capacity penalty from not knowing the channel at the receiver, unlike in the high SNR ....

S. Verdu, \On channel capacity per unit cost," IEEE Trans. Info. Theory, vol. 36, pp. 1019-30, Sept. 1990.

.... the average probability of error iff (121) 122) and (123) for some joint pmf on of the form where the time sharing random variable with values in the set is arbitrary, but may be limited to assume two values, say [44] Extensions to account for average input constraints are discussed in [66] [121], and [127] Lowcomplexity codes for the MAC are discussed in [70] and [105] It is interesting to note that even for a known MAC, the average probability of error and the maximal probability of error criteria can lead to different capacity regions [54] this is in contrast with the capacity of a ....

S. Verd u, "On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. 36, pp. 1019--1030, Sept. 1990.

....[282] the error exponent for the case of infinite bandwidth but finite power has been evaluated in the no CSI scenario. This model, where performance is measured per unit cost (power) as otherwise the system is unrestricted, is one of the few fading channel models for which the exact capacity [304] and error exponent [96] can be evaluated. In [176] the random coding error exponent has been studied for the case of multiple transmit and receive antennas and for the block fading channel. In [85] the random coding error exponent for a single dimensional fading channel is evaluated with ideal ....

S. Verd u, "On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. 36, pp. 1019--1030, Sept. 1990.

....of infinitely many degrees of freedom. Telatar [13] specializes Gallager s results to the Rayleigh fading channel. He shows that with very high bandwidths and at high SNRs, the Rayleigh fading channel has the same capacity per unit energy as an AWGN channel with the same SNR and bandwidth. Verd u[14] concentrates on capacity per unit energy cost instead of the reliability function per unit energy and generalizes Gallager s idea of capacity per unit energy to include more general cost functions. He derives a simple characterization for the capacity per unit cost for memoryless channels. He ....

....1 X i=0 OE( i ) 4) where OE( oe 2 Gamma log (1 oe 2 ) 5) 4.1. Capacity per unit energy cost In this section we derive the capacity per unit energy cost of WSSUS fading channel. Since there is only one input function, namely, u(t) j 0, having zero energy, we can apply Verd u s [14] results directly. Thus, the capacity per unit energy cost denoted by CE , is given by CE = sup u6=0 D(u) E(u) where u denotes the input waveform and E(u) R s ju(s)j 2 ds is the energy of the waveform u. We have that D(u) Tr( Sigma) oe 2 Gamma 1 X i=0 log (1 i oe 2 ....

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S. Verd'u, "On channel capacity per unit cost," IEEE Trans. Info. Th., vol. IT-34, pp. 1019--1030, Sept. 1990.

....per unit cost is given by, Er ( R) max 0ae1 ( sup u:cost(u)6=0 ( Gamma(1 ae) log E[ 1= 1 ae) jU=0] cost(u) Gamma ae R; 2) where cost(u) is the cost associated with input u, and = dP Y jU=u =dP Y jU=0 is the likelihood ratio of the input u with respect to the 0 input. Verd u[7] considered capacity per unit energy cost and showed that C cost = sup x:cost(x)6=0 D(P Y jX=x jjP Y jX=0 ) cost(x) 3) where D( Deltajj Delta) is the Kullback Liebler distance between measures. III. Capacity and Reliability Function Calculations Consider signaling over a time interval of ....

.... shown that JC (u) G 2 H R ju(t)j 4 dt and similarly that JC (u) G 2 H R jU(f)j 4 df where U(f) is the Fourier transform of u(t) We can, thereafter, show the following sup u6=0 D(u) JC (u) 1 2oe 4 : 7) Considering JC (u) as a cost function and applying the result of Verd u [7] yields the following theorem. Theorem III.1 The capacity per unit fourthegy, CJ , of the WSSUS fading channel is given by CJ = 1 2oe 4 : 8) As a consequence we have for any input random process U , I(U ; Y ) 1 2oe 4 E[JC (U ) 9) where Y is the output random process and the ....

S. Verd'u, "On channel capacity per unit cost," IEEE Trans. Info. Th., 1990, Sept., vol. IT-34, no. 5, pp. 1019-1030.

....high signal energies, the Rayleigh fading channel has the same capacity divided by energy as an AWGN channel with SNR equaling 1 and infinite bandwidth. This led him to conclude that the infinite bandwidth Rayleigh fading channel has the same capacity as the infinite bandwidth AWGN channel. Verd u [16] concentrated on capacity per unit energy instead of the reliability function per unit energy and generalized Gallager s idea of capacity per unit energy to include more general cost functions. He derived a simple characterization for the capacity per unit cost for memoryless channels. He showed ....

....upper bound to the reliability function E ffl (R) Thus, E ss (R) lim ffl 0 E ffl (Rffl 2 ) ffl 2 = E ss sl (R) E ss r (R) 2.49) and the result in (2.15) holds. 2 2.3 Capacity Per Unit Cost We briefly explain the notion of capacity per unit cost in this section. See Verd u [16] for a detailed treatment of capacity per unit cost. Consider a discrete time channel without feedback and with arbitrary input and output alphabets denoted by A and B respectively. An (N; M; fi; ffl) code is one in which the blocklength is equal to N ; the number of codewords is equal to M ; each ....

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S. Verd'u, "On channel capacity per unit cost," IEEE Trans. Info. Th., vol. IT-34, no. 5, pp. 1019--1030, Sept. 1990.

....an asymptotically optimal signaling scheme is to use only one of the subchannels (one with maximum value of C ss ) and to use antipodal signaling on that subchannel. Remark: Closely related to the capacity for small peak signal constraints is the notion of capacity per unit cost de ned by Verd u [16] with the cost of an input symbol x being the energy kxk 2 . The capacity per unit energy, CE , is the supremum over of C = 2 , where C is the capacity subject to the constraint that the average energy per symbol of each transmitted codeword be at most 2 . Moreover, the supremum ....

....x being the energy kxk 2 . The capacity per unit energy, CE , is the supremum over of C = 2 , where C is the capacity subject to the constraint that the average energy per symbol of each transmitted codeword be at most 2 . Moreover, the supremum over is achieved as 0 [16]. Every valid codeword in the de nition of C has peak energy per channel use at most , and therefore average energy per channel use at most 2 , so C C for all 0. Therefore, C ss CE . The inequality can be strict. For example, for the Rician channel CE = 2 2 ) ....

S. Verdu, \On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. IT-34, no. 5, pp. 1019-1030, Sept. 1990. 21

....[24] specialized Gallager s results to the Rayleigh fading channel and obtained the capacity divided by energy as a function of bandwidth and signal energy, concluding from this that the in nite bandwidth Rayleigh fading channel has the same capacity as the in nite bandwidth AWGN channel. Verd u [25] considered capacity per unit cost for general cost functions and derived a simple expression for the capacity per unit cost for memoryless channels for certain cost functions. 1 Kennedy [13] considered the capacity per unit time of di use WSSUS fading channels. Using an M ary ....

....a Vector Rayleigh Channel In this section the theory of capacity per unit cost is applied to derive a basic information inequality for a vector Rayleigh fading channel. 2. 1 Background: Capacity Per Unit Cost We brie y review the notion of capacity per unit cost in this section, following Verd u [25]. Consider a discrete time channel without feedback and with arbitrary input and output alphabets denoted by A and B respectively. An (N; M; code is one in which the blocklength is equal to N ; the number of codewords is equal to M ; each codeword (x m1 ; xmN ) m = 1; M , ....

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S. Verdu, \On channel capacity per unit cost," IEEE Trans. Info. Th., vol. IT-34, pp. 1019-1030, Sept. 1990.

....of T is determined by the distribution of X. The mean of T , in units of ticks per channel usage, is represented by E(T ) The mutual information in units of bits per tick I t (X; Y ) for a DMC is I t (X; Y ) I u (X; Y ) E(T ) The capacity in units of bits per tick for a DMC is given by [23] C t = max I u (X; Y ) E(T ) 1) maximized as before. Of course, if this is a constant time DMC the value E(T ) is distribution independent and we have our previous formula C t = Gamma1 Cu , where = E(T ) Note that, in general, C t is not max Iu (X;Y ) maxE(T ) see [15] If we ....

Sergio Verd'u. On channel capacity per unit cost. IEEE Transactions on Information Theory, 36(5):1019--1030, September 1990.

....distinguish them from High, then this transmission is noisy. In fact, if we look at capacity in units of bits transmission, we simply have the Zchannel [10, 4] a two symbol channel where one of the symbols is transmitted perfectly) However, the capacity in terms of bits ms is more complicated [28]. As mentioned, an important measure of the potential damage of a STC is the (channel) capacity. In our studies of STC capacity, we noticed some inconsistencies in the definition of capacity [27] We discuss this and also offer a novel and simple proof of one of the major theorems concerning the ....

.... H(X) denote the entropy of X and I(X; Y ) the mutual information (in units of bits per transmission) The mutual information in units of bits per tick for a discrete memoryless channel is I t = I(X; Y ) E(T ) 4) where E(T ) is the mean time for a symbol to be transmitted over the channel, see [25, 26, 28]. Of course, for a STC this reduces to I t = H(X) E(T ) Since the channel is memoryless, the distribution on X is stationary; this corresponds to the unconstrained symbol condition mentioned in the previous section. A rigorous study of Equation (4) for the memoryless channel in general has ....

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Sergio Verd'u. On channel capacity per unit cost. IEEE Transactions on Information Theory, 36(5):1019--1030, September 1990.

....the unfaded capacity . To obtain the spectral efficiency as a function of , 11) is used in conjunction with (10) with , since in this section we are assuming and . In the very noisy region, it is easy to see that the required for reliable communication does not change with fading, because (cf. [15]) 13) 14) 15) 1.59175 dB (16) The reason for the insensitivity of to the fading distribution lies in the fact that to achieve minimum energy per transmitted bit, ON OFF signaling is optimum (e.g. 15] 17] and the optimum receiver averages the received signal in the on periods of each of ....

....see that the required for reliable communication does not change with fading, because (cf. 15] 13) 14) 15) 1. 59175 dB (16) The reason for the insensitivity of to the fading distribution lies in the fact that to achieve minimum energy per transmitted bit, ON OFF signaling is optimum (e.g. [15] [17] and the optimum receiver averages the received signal in the on periods of each of the possible codewords and weighs each matched filter output by its corresponding fading coefficient. Thus, performance is a function of the second moment of the fading coefficients. At , the slope of the ....

S. Verd, "On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. 36, pp. 1019--1030, Sep. 1990.

....function in the setting of a binary input channel where information is normalized, not to blocklength, but to the number of s contained in the codeword. More generally, we can pose the capacity per unit cost problem where an arbitrary cost function is defined on the input alphabet [10]. An important class of cost functions are those which, like energy, assign a zero cost to one of the input symbols. For those cost functions, the capacity per unit cost not only is equal to the derivative at zero cost of the Shannon capacity but admits a simple formula [10] Even in this more ....

....on the input alphabet [10] An important class of cost functions are those which, like energy, assign a zero cost to one of the input symbols. For those cost functions, the capacity per unit cost not only is equal to the derivative at zero cost of the Shannon capacity but admits a simple formula [10]. Even in this more general setting, capacity per unit cost is achieved by on off signaling with vanishing duty cycle. A wide variety of digital communication systems (particularly in wireless, satellite, deep space, and sensor networks) operate in the power limited region where both spectral ....

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S. Verd, "On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. 36, pp. 1019--1030, Sept. 1990.

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S. Verd, "On the channel capacity per unit cost," IEEE Transactions on Information Theory, vol. 36, pp. 1361--1391, September 1990.

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S. Verd, "On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. 36, pp. 1019--1030, Sept. 1990.

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S. Verd u, "On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. 36, pp. 1019-1030, Sept. 1990.

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S. Verd u, "On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. 36, pp. 1019-1030, Sept. 1990.

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S. Verd u, "On channel capacity per unit cost ," IEEE Trans. on Information Theory, vol. 36, pp. 1019--1030, September 1990.

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S. Verdu, \On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. IT-36, no. 5, pp. 1019-1030, Sept. 1990.

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S. Verd, "On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. 36, pp. 1019--1030, Sept. 1990.

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S. Verd'u, "On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. 36, pp. 1019-1030,

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S. Verd'u, "On channel capacity per unit cost," IEEE Trans. Inform. Theory, vol. 36, pp. 1019--1030, Sept. 1990.

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