Question

A siren is emitting a constant frequency in a noisy room, $x(t) = \cos(10t)$. There is a microphone in the room measuring the signal, $x(t)$, plus the background noise denoted as the signal, $e(t)$. Let the noise signal be $g(t) = x(t) + e(t)$. The microphone signal is connected to an A/D converter which samples the signal at a rate of 55 rad/sec. You take 323 samples from the A/D converter and calculate the DFT. Recall that $x(t) = \cos(\Omega_0 t) = \frac{1}{2} (e^{j\Omega_0 t} + e^{-j\Omega_0 t})$ What integer index, $k$, of the DFT, $X[k]$, corresponds to the frequency closest to the complex component $\frac{1}{2} e^{j\Omega_0 t}$ present in the input signal? Provide your answer as an integer value.

          A siren is emitting a constant frequency in a noisy room, $x(t) = \cos(10t)$. There is a microphone in the room measuring the signal, $x(t)$, plus the background noise denoted as the signal, $e(t)$. Let the noise signal be $g(t) = x(t) + e(t)$. The microphone signal is connected to an A/D converter which samples the signal at a rate of 55 rad/sec. You take 323 samples from the A/D converter and calculate the DFT. Recall that
$x(t) = \cos(\Omega_0 t) = \frac{1}{2} (e^{j\Omega_0 t} + e^{-j\Omega_0 t})$
What integer index, $k$, of the DFT, $X[k]$, corresponds to the frequency closest to the complex component $\frac{1}{2} e^{j\Omega_0 t}$ present in the input signal?
Provide your answer as an integer value.

A siren is emitting a constant frequency in a noisy room, x(t) = cos(10t). There is a microphone in the room measuring the signal, x(t), plus the background noise denoted as the signal, e(t). Let the noise signal be g(t) = x(t) + e(t). The microphone signal is connected to an A/D converter which samples the signal at a rate of 55 rad/sec. You take 323 samples from the A/D converter and calculate the DFT. Recall that
x(t) = cos(Ω0 t) = (1)/(2) (e^jΩ0 t + e^-jΩ0 t)
What integer index, k, of the DFT, X[k], corresponds to the frequency closest to the complex component (1)/(2) e^jΩ0 t present in the input signal?
Provide your answer as an integer value.

Added by John D.

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