If input signals are delayed ,Then autocorrelation of delayed input signal is same as that of autocorrelation of original signal.Select one:FALSECan't sayTRUE

Step 1: Break Down the Problem

We need to evaluate whether the autocorrelation of a delayed signal is the same as the autocorrelation of the original signal. Specifically, we will analyze the properties of autocorrelation regarding delays.

Step 2: Relevant Concepts

The autocorrelation function $R_x(\tau)$ of a signal $x(t)$ is defined as: $R_x(\tau) = \int_{-\infty}^{\infty} x(t) x(t+\tau) \, dt$ If we delay the signal by $T$, the delayed signal $x(t - T)$ will have an autocorrelation defined as: $R_{x_T}(\tau) = \int_{-\infty}^{\infty} x(t - T) x(t - T + \tau) \, dt$

Step 3: Analysis and Detail

To see if these two autocorrelation functions are the same, we can make a substitution in the delayed autocorrelation computation:

1. Let $u = t - T$, then $dt = du$, and the limits of integration remain the same.
2. Therefore: $R_{x_T}(\tau) = \int_{-\infty}^{\infty} x(u) x(u + \tau) \, du = R_x(\tau)$ This shows that the autocorrelation of the delayed signal is equal to the autocorrelation of the original signal regardless of the delay $T$.

Step 4: Verify and Summarize

We verified that $R_{x_T}(\tau) = R_x(\tau)$ holds true for any delay $T$. Thus, the statement can be concluded.

Final Answer

TRUE: The autocorrelation of the delayed input signal is the same as that of the original signal.