To compute the inner product $(x_1(t), x_2(t))$ of the signals $x_1(t)$ and $x_2(t)$, we need to evaluate the integral:

$(x_1(t), x_2(t)) = \int_{-\infty}^{\infty} x_1(t) \cdot x_2^*(t) \, dt$

where $x_1(t) = \frac{W}{\pi} \, \text{sinc} \left( \frac{Wt}{\pi} \right)$ and $x_2(t) = \frac{W}{\pi} \, \text{sinc} \left( \frac{Wt}{\pi} \right) \sin(\omega t)$. We will also use the fact that the inner product involves the conjugate of $x_2(t)$, which means $x_2^*(t) = \frac{W}{\pi} \, \text{sinc} \left( \frac{Wt}{\pi} \right) \sin(\omega t)$ since the sinc and sine functions are real-valued.

Step-by-step Solution

1. Substitute the expressions for $x_1(t)$ and $x_2^*(t)$ into the inner product formula:

$(x_1(t), x_2(t)) = \int_{-\infty}^{\infty} \left( \frac{W}{\pi} \, \text{sinc} \left( \frac{Wt}{\pi} \right) \right) \left( \frac{W}{\pi} \, \text{sinc} \left( \frac{Wt}{\pi} \right) \sin(\omega t) \right) dt$

$= \left( \frac{W}{\pi} \right)^2 \int_{-\infty}^{\infty} \text{sinc}^2 \left( \frac{Wt}{\pi} \right) \sin(\omega t) \, dt$

2. Interpret the integral:

The integral $\int_{-\infty}^{\infty} \text{sinc}^2 \left( \frac{Wt}{\pi} \right) \sin(\omega t) \, dt$ is quite complex, but we can simplify it by recognizing the role of the frequency-shift property and Parseval's identity.

Frequency-Shift Property

The frequency-shift property states that multiplying a time-domain signal by a sinusoidal function corresponds to a shift in the frequency domain. If we denote the Fourier transform of $x_1(t)$ by $X_1(f)$, then the Fourier transform of $x_2(t)$, which is $x_1(t) \sin(\omega t)$, can be written as:

$X_2(f) = \frac{1}{2} \left[ X_1(f - \omega) + X_1(f + \omega) \right]$

The inner product in the frequency domain becomes:

$(x_1(t), x_2(t)) = \int_{-\infty}^{\infty} X_1(f) X_2^*(f) df$

Using the frequency-shift property, we can analyze this inner product in the frequency domain.

Parseval's Identity

Parseval's identity states that the inner product of two functions in the time domain is equal to the inner product of their Fourier transforms in the frequency domain:

$(x_1(t), x_2(t)) = \int_{-\infty}^{\infty} X_1(f) X_2^*(f) \, df$

Calculation

Without solving the exact integral step-by-step (which would be complex), the approach would be to use the frequency-shift property to compute the shifted Fourier transforms of $x_1(t)$ and $x_2(t)$ and then apply Parseval’s identity to evaluate the inner product in the frequency domain.

Final Result (Heuristic)

Given the complexity of the integration in the time domain, this problem is often tackled by using the frequency domain interpretation. After applying the properties mentioned, the inner product should yield a result that involves the spectral content of the two signals.

Would you like a more detailed explanation of any of these properties or a numerical approach to solve this?