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Explain Autocorrelation functions with question.
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Solution:

Autocorrelation functions Setting $u=v$, we specialize in the continuous-time and discrete-time autocorrelation functions,

$ R_u(\tau)=R_{u, u}(\tau)=\int u(t) u^*(t-\tau) \mathrm{d} t,\\ $

$ R_u[n]=R_{u, u}[n]=\sum_l u[l] u^*[l-n] .\\ $

We now express the continuous-time crosscorrelation function in terms of the discrete-time crosscorrelation function for the particular case of a rectangular chip waveform time-limited to an interval of length $T_{\mathrm{c}}$.

$ \begin{aligned}\\ R_{u, v}(\tau) &=\sum_l \sum_k u[l] v^*[k] \int \psi\left(t-l T_{\mathrm{c}}\right) \psi^*\left(t-k T_{\mathrm{c}}-\tau\right) \mathrm{d} t \\\\ &=\sum_l \sum_k u[l] v^*[k] r_\psi\left((k-l) T_{\mathrm{c}}+\tau\right) .\\ \end{aligned}\\ $

Setting $\tau=(D+\delta) T_{\mathrm{c}}$ as before, where $D=\left\lfloor\tau / T_{\mathrm{c}}\right\rfloor$ is an integer, and $\delta \in[0,1)$, we get,

$ R_{u, v}(\tau)=\sum_{l, k} u[l] v^*[k] r_\psi\left((k+D-l) T_{\mathrm{c}}+\delta T_{\mathrm{c}}\right) .\\ $

For the rectangular chip waveform, we have,

$ r_\psi\left((k+D-l) T_{\mathrm{c}}+\delta T_{\mathrm{c}}\right)= \begin{cases}1-\delta, & k+D-l=0, \\\\ \delta, & k+D-l=-1, \\\\ 0, & \text { else. }\end{cases}\\ $

This means that the only nonzero terms in (8.67) correspond to $k=l-D$ and $k=l-D-1$. Substituting (8.68) into (8.67), we obtain,

$ \begin{aligned}\\ R_{u, v}(\tau) &=(1-\delta)\left(\sum_l u[l] v^*[l-D]\right)+\delta\left(\sum_l u[l] v^*[l-D-1]\right) \\\\ &=(1-\delta) R_{u, v}[D]+\delta R_{u, v}[D+1] .\\ \end{aligned}\\ $

The preceding expression shows that the continuous-time crosscorrelation function can be made small for an arbitrary delay $\tau$ by making the discrete-time crosscorrelation function small (on average) for all integer delays different from zero.

By specializing to $u=v$, we see that this observation also holds for autocorrelation functions.

While these conclusions are based on a rectangular timelimited chip waveform, Problem 8.20 generalizes to arbitrary chip waveforms $\psi$.

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