Correct Expression of Carrier Frequency Offset Estimate ... - IEEE Xplore

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 10, OCTOBER 2010

3039

Correct Expression of Carrier Frequency Offset Estimate in Fully-Loaded OFDM Systems Qi Cheng, Member, IEEE

Abstract—The expression of an estimate of carrier frequency offset in a paper published on the June 2009 issue of this journal, is flawed. A correct expression is presented. Index Terms—Carrier frequency offset, fully-loaded OFDM system.

F

OR fully-loaded OFDM systems, the cost function for carrier frequency offset (CFO) estimation is given in the following form [1] 𝑓 (𝑢) = 𝐶1 + 𝐶2 cos(2𝜋𝑢 + 𝐶3 )

(1)

where 𝑢 (∈ (−0.5, 0.5]) is a free variable for the CFO denoted by 𝜖, 𝐶𝑖 ’s for 𝑖 = 1, 2, 3 are independent of 𝑢 and can be calculated from noisy measurement. 𝐶2 ∕= 0 is required to guarantee a unique solution. An estimate of the CFO is determined from (1) as a minimizing argument. Given values of 𝑓 (0), 𝑓 (1/2), 𝑓 (1/4), one can compute 𝑎 = [𝑓 (0) + 𝑓 (1/2)]/2 − 𝑓 (1/4)

(2)

𝑏 = [𝑓 (0) − 𝑓 (1/2)]/2.

(3)

In terms of 𝑎 and 𝑏, (13) of [1] expression for the CFO estimate ⎧ − arctan(𝑎/𝑏)/(2𝜋)   ⎨ − arctan(𝑎/𝑏)/(2𝜋) 𝜖ˆ = −(𝜋 + arctan(𝑎/𝑏))/(2𝜋)   ⎩ −(−𝜋 + arctan(𝑎/𝑏))/(2𝜋)

presents the following if if if if

𝑎≥0 𝑎