Question:hard
Two analog signals \(x_1(t)\) and \(x_2(t)\) (\(t\) in second) are sampled at a rate \(F_s=40\) Hz, where
\[ x_1(t)=\cos(20\pi t),\ t\geq0,\qquad x_2(t)=\cos(100\pi t),\ t\geq0. \] The first ten samples (starting from \(t=0\)) are considered for the analysis. Which of the following statements is TRUE?
Two analog signals \(x_1(t)\) and \(x_2(t)\) (\(t\) in second) are sampled at a rate \(F_s=40\) Hz, where
\[ x_1(t)=\cos(20\pi t),\ t\geq0,\qquad x_2(t)=\cos(100\pi t),\ t\geq0. \] The first ten samples (starting from \(t=0\)) are considered for the analysis. Which of the following statements is TRUE?
\[ x_1(t)=\cos(20\pi t),\ t\geq0,\qquad x_2(t)=\cos(100\pi t),\ t\geq0. \] The first ten samples (starting from \(t=0\)) are considered for the analysis. Which of the following statements is TRUE?
Show Hint
Sample both cosines at t_n = n/40 seconds and simplify cos(2.5*pi*n) using the fact that 2*pi*n is always a whole number of cycles.
Updated On: Jul 20, 2026
- All of the first three samples of \(x_1(t)\) are greater than the corresponding samples of \(x_2(t)\).
- All of the last three samples of \(x_1(t)\) are greater than the corresponding samples of \(x_2(t)\).
- All of the samples of \(x_2(t)\) are greater than the corresponding samples of \(x_1(t)\).
- All of the fourth to seventh samples of \(x_1(t)\) are equal to the corresponding samples of \(x_2(t)\).
Show Solution
The Correct Option is D
Solution and Explanation
Step 1: Read off the two signal frequencies.
$x_1(t)=\cos(20\pi t)$ has frequency $f_1=10$ Hz, since $2\pi f_1=20\pi$. Likewise $x_2(t)=\cos(100\pi t)$ has frequency $f_2=50$ Hz.
Step 2: Find the alias frequency of $x_2$ under 40 Hz sampling.
Sampling at $F_s=40$ Hz can only distinguish frequencies up to the Nyquist limit of $20$ Hz; anything above that folds back. The alias of $50$ Hz is found by subtracting the sampling rate until the result lands inside $[0,F_s/2]$:
\[ f_{2,alias}=50-40=10\text{ Hz} \] This lands exactly on $10$ Hz, with no extra folding needed since $10<20$.
Step 3: Conclude about the samples.
Since the alias frequency of $x_2$ exactly equals the frequency of $x_1$, and both are plain cosines with zero phase, the two sampled sequences $x_1[n]$ and $x_2[n]$ are identical at every sample index $n$, not just approximately close.
Step 4: Compute the actual values to check the options.
With $T_s=1/40$ s, $x_1[n]=\cos(\pi n/2)$ gives the ten values $1,0,-1,0,1,0,-1,0,1,0$ for $n=0$ to $9$, and $x_2[n]$ gives the identical list. So any statement claiming one sequence is strictly greater than the other anywhere is false, while a statement claiming a block of samples are equal, such as the fourth to seventh, is true.
\[ \boxed{\text{All of the fourth to seventh samples of }x_1(t)\text{ are equal to the corresponding samples of }x_2(t).} \]
$x_1(t)=\cos(20\pi t)$ has frequency $f_1=10$ Hz, since $2\pi f_1=20\pi$. Likewise $x_2(t)=\cos(100\pi t)$ has frequency $f_2=50$ Hz.
Step 2: Find the alias frequency of $x_2$ under 40 Hz sampling.
Sampling at $F_s=40$ Hz can only distinguish frequencies up to the Nyquist limit of $20$ Hz; anything above that folds back. The alias of $50$ Hz is found by subtracting the sampling rate until the result lands inside $[0,F_s/2]$:
\[ f_{2,alias}=50-40=10\text{ Hz} \] This lands exactly on $10$ Hz, with no extra folding needed since $10<20$.
Step 3: Conclude about the samples.
Since the alias frequency of $x_2$ exactly equals the frequency of $x_1$, and both are plain cosines with zero phase, the two sampled sequences $x_1[n]$ and $x_2[n]$ are identical at every sample index $n$, not just approximately close.
Step 4: Compute the actual values to check the options.
With $T_s=1/40$ s, $x_1[n]=\cos(\pi n/2)$ gives the ten values $1,0,-1,0,1,0,-1,0,1,0$ for $n=0$ to $9$, and $x_2[n]$ gives the identical list. So any statement claiming one sequence is strictly greater than the other anywhere is false, while a statement claiming a block of samples are equal, such as the fourth to seventh, is true.
\[ \boxed{\text{All of the fourth to seventh samples of }x_1(t)\text{ are equal to the corresponding samples of }x_2(t).} \]
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