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If the residual number of sensing periods overlapped with a primary signal until the primary signal disappears is n (say that a secondary user has n detection opportunities), then the probability P d,i that a secondary user detects the primary signal at its i th detection opportunity is P d, i = { p d 1 − p d i − 1, if 1 ≤ i ≤ n, 0, otherwise ( n = i = 0 ).
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The probability that a secondary user detects the primary signal within n detection opportunities, P d (n), is given by P d n = ∑ i = 0 n P d, i = 1 − 1 − p d n. (4).
Since we are interested in the detection probability within TRDT, the number of detection opportunities is upper-bounded.
We first derive the upper-bounded number of detection opportunities for the CID case (denoted by n ¯ CID T I ).
Moreover, the upper-bounded number of detection opportunities for the CID case differs according to the range of RIDLE.
(3) Figure 2 Illustration of IMD case and the example of upper-bounded number of detection opportunities in IMD case.
We can observe that every curve sharply drops around the region where TI = TRDT, due to the decrement of the upper-bounded number of detection opportunities in (6) and (10).
The probability that a secondary user has n detection opportunities with TI can be given by P DO IMD n, T I = Pr R BUSY ≥ n T I − Pr R BUS Y ≥ n + 1 T I. (5).
The upper-bounded number of detection opportunities for the IMD case can be expressed as a function of TI as n ¯ IMD T I = T RDT T I, (6).
To derive the number of detection opportunities under kTI ≤ RIDLE < (k + 1 TI, it is required that a secondary user does not commit any false alarms during the fully idle intervals (i.e., k sensing intervals).
e.g., in Figure 3a, n ¯ CID T I = n ¯ CID − T I = 2 and in Figure 3b, n ¯ CID T I = n ¯ CID + T I = 3. Accordingly, the probability that a secondary user has n detection opportunities with TI differs depending on the range of RIDLE.
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