1.11 Ideal reconstruction of sampled signals

Reconstruction of sampled signals

How do we go from $x(n)$ to CT ( )?

Step 1

Place $x(n)$ into CT on an impulse train $s(t)$ ( ).

Step 2

Pass $x_s(t)$ through an idea lowpass filter $H_{\mathrm{LP}}()$ ( ).

If we had no aliasing then $x_r(t)=x_c(t)$ , where $x(n)=x_c(nT)$ .

Ideal reconstruction system

In frequency domain:

• $$X_s()=X(T)$$ where $X(T)$ is the DTFT of $x(n)$ at digital frequency $=T$ .
• $$X_r()=H_{\mathrm{LP}}()X_s()$$
  $$X_r()=H_{\mathrm{LP}}()X(T)$$

In time domain:

• $$x_s(t)=\sum$$∞ ∞ x n t n T
• $$x_r(t)=(\sum )$$∞ ∞ x n t n T h LP t $$h_{\mathrm{LP}}(t)=\mathrm{sinc}(\frac{\pi }{T}t)$$