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...T OF THEOREM 3.2 First, we observe that the rate region specified in Theorem 3.2 is a convex set in Proposition D.1. We will use the following standard result from convex analysis (see, for instance, =-=[17]-=-) in the proof of the converse. Proposition A.1: A point is on the boundary of the a capacity region if and only if there exists a such that the linear functional achieves its maximum, over all in the...

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...result we will consider this standard form of the cognitive channel without loss of generality and we will refer to it as the cognitive (1, a, b, 1) channel. 4 These transformations were used in [1], =-=[3]-=- and [16], in the context of the classical interference channel. 5sPrimary radio X n p (mp) X n c (mc, mp) Cognitive radio 1 1 a b Z n p + + Z n s Primary receiver Y n p Y n s Secondary receiver Figur...

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...he property that the primary decoder does not depend on the encoder and decoder used by the secondary system. This cooperative communication scenario can be thought of as an interference channel [1], =-=[16]-=-, [4] but with degraded message sets 2 : Achievable schemes for this channel have been first studied in [6]. A related problem of communicating a single private message along with a common message to ...

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...X + � Zs. (89) The new noise vectors are given by � Zp ∼ N (0, H −1 p ΣzH −T p ) and � Zs ∼ N (0, H −1 s ΣzH −T s ). This channel is then exactly in the form of an Aligned MIMO BC channel (AMBC) (see =-=[19]-=-, Section 2), but with mp revealed to the secondary receiver. Let � Y n p ∈ R 2×n and � Y n s ∈ R 2×n denote the channel outputs over a block of n channel uses. We can upper bound any achievable rate ...

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...i)P(m = i), (126) n 2 log � e 2 n h(Y n 2 |m=i) + e n h(Zn � ) P(m = i), (127) � e 2 n h(Y n 2 |m) + e n h(Zn � ) , (128) where (a) follows from the classical Entropy Power Inequality (EPI) (see e.g. =-=[5]-=-), and (b) follows from Jensen’s inequality applied to the convex function log(e 2x/n + k) (for constant k and n). �. Lemma D.1 Given two zero-mean random variables X and Y with a fixed covariance mat...

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...has the property that the primary decoder does not depend on the encoder and decoder used by the secondary system. This cooperative communication scenario can be thought of as an interference channel =-=[1]-=-, [16], [4] but with degraded message sets 2 : Achievable schemes for this channel have been first studied in [6]. A related problem of communicating a single private message along with a common messa...

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...rt of Theorem 4.1 First we observe that the rate-region specified in Theorem 4.1 is a convex set in Proposition D.1. We will use the following standard result from convex analysis (see, for instance, =-=[15]-=-) in the proof of the converse. Proposition A.1 A point R ∗ = (R ∗ p, R ∗ c) is on the boundary of the a capacity region if and only if there exists a µ ≥ 0 such that the linear functional µRp + Rc ac...

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...user’s rate is unaffected. 5.1.2 Robustness to noise statistics All our results have been derived under the assumption that the noise affecting the receivers, Z n p and Z n s , is i.i.d. Gaussian. In =-=[2]-=- it was shown that using a Costa encoderdecoder pair that is designed for additive i.i.d Gaussian noise on a channel with arbitrary (additive) noise statistics will cause no loss in the achievable rat...

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...does not depend on the encoder and decoder used by the secondary system. Under such a joint encoder/decoder design, the cognitive radio channel can be thought of as the classical interference channel =-=[1]-=-, [18], [4] but with degraded message sets2: Achievable schemes for this channel have been first studied in [6]. The capacity region of this channel, in the low-interference-gain regime, has recently ...

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...perty that the primary decoder does not depend on the encoder and decoder used by the secondary system. This cooperative communication scenario can be thought of as an interference channel [1], [16], =-=[4]-=- but with degraded message sets 2 : Achievable schemes for this channel have been first studied in [6]. A related problem of communicating a single private message along with a common message to each ...

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...rence” regime in which the cognitive radio is closer to its receiver than to the primary receiver. In this regime, our results are subsumed by the results derived in a concurrent and independent work =-=[24]-=-. We also demonstrate that, in a “high interference” regime, multi-user decoding at the primary receiver is optimal from the standpoint of maximal jointly achievable rates for the primary and cognitiv...

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...ive i.i.d Gaussian noise on a channel with arbitrary (additive) noise statistics will cause no loss in the achievable rates. 7 Combined with the similar classical result for the standard AWGN channel =-=[11]-=-, we see that the maximal rate expressed in Theorem 3.1 is achievable for all noise distributions. 6 Classical RTS/CTS solutions to this problem are not viable since they require that the primary syst...

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...ms plausible that the cognitive radio could decode 9 the message of the primary user in fewer channel uses than are required by the primary receiver. Recent work in distributed space-time code design =-=[13]-=- indicates that this overhead decoding delay is negligible if the cognitive radio has as little as a 10 dB advantage in the received SNR over the primary receiver. 5.3 Extension to complex baseband Th...

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... + 1)) (55) (e) ≤ n 2 log (2πe ((1 − α)Pc + 1)) , (56) where (a) follows from the chain rule and (b) follows from the fact that conditioning does not increase entropy, and (c) follows from Lemma 1 of =-=[19]-=- (a more direct proof appears in Lemma 1 of [18]). Equality (d) follows from the following argument: Since jointly Gaussian Xp,j, � Yp,j achieve equality in (c) (by Lemma 1 of [19]), we can without lo...

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...h the incumbent licensed (primary) users, generating minimal interference while somehow taking advantage of the available resources. Such devices could, for instance, form real-time secondary markets =-=[14]-=- for the licensed spectrum holders of a cellular network or even, potentially, allow a complete secondary system to simultaneously operate in the same frequency band as the primary. ∗ A. Jovičić and P...

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...on. 25where (a) follows from the chain rule and (b) follows from the fact that conditioning does not increase entropy, and (c) follows from Lemma 1 of [20] (a more direct proof appears in Lemma 1 of =-=[19]-=-). Equality (d) follows from the following argument: Since jointly Gaussian Xp,j, ˜ Yp,j achieve equality in (c) (by Lemma 1 of [20]), we can without loss of generality, let √ Xc,j = ˆ Xc,j + Pc,j αj ...

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...e provide an alternative proof for a portion of the capacity region of the same channel. The capacity region under the assumption that both receivers experience high interference has been reported in =-=[13]-=-. A related problem of communicating a single private message along with a common message to each of the receivers has been studied in [14]. In contrast to our results for the low-interference-gain re...

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... receivers experience high interference has been reported in [13]. A related problem of communicating a single private message along with a common message to each of the receivers has been studied in =-=[14]-=-. In contrast to our results for the low-interference-gain regime, we exhibit a regime in which joint code design is beneficial with respect to the largest set of simultaneously achievable rates of th...

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...ognitive nature of such devices, we assume that the cognitive radio has non-causal knowledge of the codeword of the primary user in its vicinity 1 ; in this, we are motivated by the model proposed in =-=[6]-=-. We address the following fundamental question: What is the largest rate that the cognitive radio can achieve under the constraint that (i) it generates no interference for the primary user in its vi...

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...evable schemes for this channel have been first studied in [6]. The capacity region of this channel, in the low interference gain regime, has recently been found by an independent and concurrent work =-=[23]-=-, [24]. In this paper, we provide an alternative proof for a portion of the capacity region of the same channel. The capacity region under the assumption that both receivers experience high interferen...

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...chievable schemes for this channel have been first studied in [6]. A related problem of communicating a single private message along with a common message to each of the receivers has been studied in =-=[12]-=-. Furthermore, we exhibit a regime in which joint code design is beneficial when one considers the largest set of simultaneously achievable rates of the primary and cognitive users. We show that, unli...

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...ded message sets 2 : Achievable schemes for this channel have been first studied in [6]. The capacity region under the assumption that both receivers experience high interference has been reported in =-=[12]-=-. A related problem of communicating a single private message along with a common message to each of the receivers has been studied in [13]. Recently, a concurrent and independent work [22] has obtain...