Deterministic Sampling and Reconstruction

x(t) bandlimited to $\pm W Hz$
Sample at at least 1/2W second intervals and x(t) can be obtained from samples.WHY?

no aliasing

$$\frac{1}{T}\ge 2W\rightarrow T \le \frac{1}{2W} \mbox{\qquad NYQUIST RATE}$$

aside

$$x(t)s(t)\Rightarrow X(f)\ast S(f)$$

$$s(t)=\sum_n\delta(t-nT) \mbox{is periodic so t has FS}$$

Therefore,

$$S(f)=\sum_k s_k\delta(f-kf_0)$$

$$s_k=\frac{1}{T}\int_{-T/2}^{T/2}\delta(t)e^{-j\frac{2\pi}{T}kt}dt=\frac{1}{T}$$

$$S(f)=\frac{1}{T}\sum_k\delta(f-\frac{k}{T})$$

Thererfore

$$X(f)\ast S(f)=\frac{1}{T}\sum_kX(f-\frac{k}{T})$$

Recovery $\Rightarrow$ *******FIG.7*********