1. **Problem Statement:** We are given a strictly bandlimited signal $x_a(t)$ with Fourier transform $X_a(j\Omega)$ such that $X_a(j\Omega) = 0$ for $|\Omega| > \Omega_0$. We want to understand the effects of sampling this signal at frequency $\Omega_s$ and the resulting spectra, particularly focusing on aliasing and the Nyquist sampling theorem. 2. **Strict Bandlimitedness:** The signal $x_a(t)$ has no frequency components outside the band $[-\Omega_0, \Omega_0]$. This means its spectrum $X_a(j\Omega)$ looks like a triangular shape centered at zero frequency with base from $-\Omega_0$ to $\Omega_0$. 3. **Sampling at Frequency $\Omega_s$:** Sampling $x_a(t)$ at intervals $T_s = \frac{2\pi}{\Omega_s}$ creates a sampled signal $x_s(t)$. The Fourier transform $X_s(j\Omega)$ of the sampled signal is a sum of shifted copies of $X_a(j\Omega)$ spaced by multiples of $\Omega_s$: $$X_s(j\Omega) = \frac{1}{T_s} \sum_{k=-\infty}^\infty X_a(j(\Omega - k\Omega_s))$$ 4. **Case 1: $\Omega_s \geq 2 \Omega_0$ (No Aliasing).** When the sampling frequency is at least twice the highest frequency $\Omega_0$ (the Nyquist rate), the shifted spectra copies do not overlap. Graphically this is represented by triangular spectra centered at ... $-\Omega_s$, 0, $\Omega_s$ ..., separated so they do not interfere. Because there is no overlap, the original bandlimited signal can be perfectly reconstructed using a low-pass filter with cutoff $\Omega_0$. 5. **Case 2: $\Omega_s < 2 \Omega_0$ (Aliasing Occurs).** If the sampling frequency is less than twice the highest frequency component, the shifted spectra overlap. This overlap causes spectral components from one copy to interfere with others, corrupting the frequency content. This effect is aliasing. Mathematically, the sum in step 3 produces overlapping nonzero contributions. 6. **Nyquist Sampling Theorem:** - If $x_a(t)$ is strictly bandlimited with $X_a(j\Omega) = 0$ for $|\Omega| > \Omega_0$, - And the sampling frequency satisfies $\Omega_s = \frac{2\pi}{T_s} \geq 2 \Omega_0$, - Then $x_a(t)$ can be uniquely recovered from its samples $x_a(nT_s)$ by low-pass filtering. 7. **Practical Note:** Real-world signals are never perfectly bandlimited. Therefore, an analog anti-aliasing filter is used before sampling to reduce components above the Nyquist frequency, minimizing aliasing during analog-to-digital conversion. **Final Answer: The Nyquist Sampling Theorem states that a bandlimited signal can be perfectly reconstructed from samples if the sampling frequency $\Omega_s$ is at least twice the maximum signal frequency $\Omega_0$. If $\Omega_s < 2\Omega_0$, aliasing occurs and exact reconstruction is impossible.** **Source:** - Alan V. Oppenheim and Ronald W. Schafer, "Discrete-Time Signal Processing," 3rd Edition, Prentice Hall, 2009. - Simon Haykin and Barry Van Veen, "Signals and Systems," 2nd Edition, Wiley, 2003.