Space/time/frequency methods in adaptive radar - New Jersey ...

452.5 Time-Frequency Analysis in NoiseThe discrete chirp signal in the presence of additive noise is given by:is the chirp phase, and v(n) are independent samples of acomplex white Gaussian noise process with zero mean and variance a 2 . The inputSNR is given by SNR = A²/σ² . From (2.40), the WVD kernel can be written as:In the expression above, the noise component isgiven by:The resultant noise power in the kernel isThere are only N/2 independent samples in the kernel (since it is conjugatesymmetric), so the variance of the noise at the output of the WVD is scaledby a factor of N/2 rather than N. Therefore, the SNR of the WVD output is:Equation 2.49 can be used to computea bound on the chirp estimation error. The performance of the WVD is comparedagainst the Cramér-Rao bound (CRB) on the IF estimation variance for a a complexsingle tone in additive white Gaussian noise. The CRB is achieved by the DiscreteFourier transform (DFT) of the single tone signal and is given by [45]: