On the Energy Efficiency of Orthogonal Signaling

Energy efficient transmission is of paramount importance in many communication systems and particularly in mobile wireless systems due to the scarcity of energy resources. Energy efficiency can be measured by the energy required to send one information bit reliably. It is well-known that for Gaussian channels subject to average input power constraints, the minimum received bit energy normalized by the noise spectral level is E b N0 min = -1.59 dB regardless of the availability of channel side information (CSI) at the receiver (see e.g., [1] - [7]). Golay [1] showed that this minimum bit energy can be achieved in the additive white Gaussian noise (AWGN) channel by pulse position modulation (PPM) with vanishing duty cycle when the receiver employs threshold detection. Indeed, Turin [2] proved that any orthogonal M -ary modulation scheme with envelope detection at the receiver achieves the normalized bit energy of -1.59 dB in the AWGN channel as M → ∞. It is further shown in [3] and [4] that M -ary orthogonal frequency-shift keying (FSK) achieves this minimum bit energy asymptotically as M → ∞ also in noncoherent fading channels where neither the receiver nor the transmitter knows the fading coefficients. As also well-known by now in the digital communications literature [14], these results are shown by proving that the error probabilities of orthogonal signaling can be made arbitrarily small as M → ∞ as long as the normalized bit energy (or equivalently SNR per bit) is greater that -1.59 dB. These studies demonstrate that asymptotically orthogonal signaling is optimally energy efficient and highly resilient to fading even when the receiver performs hard-decision detection. However, these asymptotical performances require operation in the infinite bandwidth regime (when M → ∞) or signaling with unbounded peak-to-average power ratio, i.e., vanishing duty cycle.

In this paper, motivated by practical considerations and limitations, we analyze the non-asymptotic energy efficiency of orthogonal signaling. We consider M -ary FSK with finite M . 1 This work was supported in part by the NSF CAREER Grant CCF-0546384.

Additionally, we investigate the energy efficiency when FSK is combined with on-off keying (or equivalently PPM) whose duty cycle is small but nonzero. On-off FSK (OOFSK) modulation introduces peakedness in both time and frequency, and provides improvements in energy efficiency over on-off keying only or FSK only.

II. CHANNEL MODEL We consider the following channel model

where x k is the discrete input, s x k is the transmitted signal when the input is x k , and r k is the received signal during the k th symbol duration. h k is the channel gain. h k is a fixed constant in unfaded AWGN channels, while in flat fading channels, h k denotes the fading coefficient. {n k } is a sequence of independent and identically distributed (i.i.d.) zero-mean circularly symmetric Gaussian random vectors with covariance matrix E{nn † } = N 0 I where I denotes the identity matrix. We assume that the system has an average energy constraint of E{ s x k2 } ≤ E ∀k. If M -ary orthogonal FSK modulation is used for transmission, then x k ∈ {1, 2, . . . , M } and the transmitted signal has an M complex-dimensional vector representation. If x k = m, s x k = s m = (s m,1 , s m,2 , . . . , s m,M ) where s m,m = √ Ee jθm and s m,i = 0 for all i = m. The phases θ m can be arbitrary. The received signal r k and noise n k are also M dimensional. We assume that the receiver quantizes the received vector r k by performing energy detection.

III. ENERGY EFFICIENCY OF FSK MODULATION In this section, we assume that the transmitter employs FSK modulation and the receiver performs energy detection. In the well-known noncoherent detection of FSK signals, s i is declared as the detected signal if the i th component of the received vector r has the largest energy, i.e., |r i | 2 > |r j | 2 ∀j = i. Note that this decision rule is the maximum likelihood decision rule in AWGN, coherent fading, and noncoherent Rician fading channels [9], [13], [14]. The output of the detector is denoted by y ∈ {1, 2, . . . , M }. Note that with energy detection, the channel can be now regarded as a symmetric discrete channel with M inputs and M outputs.

Initially, we consider the AWGN channel and assume h k = 1 ∀k. The capacity which is achieved by equiprobable FSK signals is given by 2

where SNR = E N0 and P l,1 = P (y = l|x = 1) is the probability that y = l given that x = 1. Using the results on the error probabilities of noncoherent detection of FSK signals (see e.g., [14]), we have

Hence, the capacity can also be expressed as

Next, we provide the behavior of the capacity in the low-SNR regime.

Proposition 1: The first derivative at zero SNR of the capacity C M (SNR) in ( 2) is ĊM (0) = 0. Therefore, the bit energy required at zero spectral efficiency is

Proof : From (3), we can write

Above, Ṗ1,1 (SNR = 0) denotes the derivative of the transition probabili

…(Full text truncated)…

This content is AI-processed based on ArXiv data.