Principles of Modern Radar - Volume 2 1891121537

2.2 Stretch Processing 35is applied in the signal processor to compute the spectrum and is a sampled version of thediscrete time Fourier transform (DTFT),Y (ω) =N−1∑n=0y (n) exp (− jnω) (2.29)where ω is continuous. Sampling the received signal in equation (2.13) produces(y (n) = exp j2π β )τ t d (nT S ) exp (− jθ) n = 0, 1,...,(N − 1) (2.30)where T s is the ADC sampling period, and F s = 1/T s . If no oversweep is employed, thenumber of samples, N, collected from a specific scatterer is a function of the scatterer’srelative position within the receive window[ (N = F s τ 1 − |t )]d|(2.31)τand if an oversweep of the oscillator is employed, the number of samples is independentof time delayN = [F s τ] (2.32)Applying the DTFT to the sampled signal in equation (2.30), the magnitude of the responseis( (N sin ω − 2π ))βt dY M (ω) =2 F s τ( (1∣ sin ω − 2π ))βt d(2.33)∣2 F s τThe response in equation (2.33) is known as a digital sinc function or Dirichlet function.The spectrum is periodic in 2π, and a single period is defined over −π ≤ ω