A siren is emitting a constant frequency in a noisy room, a(t) = cos(6t). There is a micropho the room measuring the signal, x(t), plus the background noise denoted as the signal, e(t). Le noise signal be g(t) = x(t) + e(t). The microphone signal is connected to an A/D converter samples the signal at a rate of 48 rad/sec. You take 193 samples from the A/D converter and calculate the DFT. Recall that x(t) = cos(Not) = (einot + e-Not) What integer index, k, of the DFT, X[k], corresponds to the frequency closest to the complex componentenot present in the input signal? Provide your answer as an integer value.
A siren is emitting a constant frequency in a noisy room, a(t) = cos(6t). There is a micropho the room measuring the signal, x(t), plus the background noise denoted as the signal, e(t). Le noise signal be g(t) = x(t) + e(t). The microphone signal is connected to an A/D converter samples the signal at a rate of 48 rad/sec. You take 193 samples from the A/D converter and calculate the DFT. Recall that x(t) = cos(Not) = (einot + e-Not) What integer index, k, of the DFT, X[k], corresponds to the frequency closest to the complex componentenot present in the input signal? Provide your answer as an integer value.
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![A siren is emitting a constant frequency in a noisy room, \( x(t) = \cos(\Theta_0 t) \). 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 48 rad/sec. You take 193 samples from the A/D converter and calculate the DFT. Recall that
\[
x(t) = \cos(\Theta_0 t) = \frac{1}{2} \left(e^{j\Theta_0 t} + e^{-j\Theta_0 t} \right)
\]
What integer index, \( k \), of the DFT, \( X[k] \), corresponds to the frequency closest to the complex component \( \frac{1}{2} e^{j\Theta_0 t} \) present in the input signal?
**Provide your answer as an integer value.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9cefd9b-1e12-4ab9-bddf-7966a130d3b6%2F8de88cfa-cc32-4991-b0f8-a1820471fb07%2Fmrwvt8u8_processed.png&w=3840&q=75)
Transcribed Image Text:A siren is emitting a constant frequency in a noisy room, \( x(t) = \cos(\Theta_0 t) \). 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 48 rad/sec. You take 193 samples from the A/D converter and calculate the DFT. Recall that
\[
x(t) = \cos(\Theta_0 t) = \frac{1}{2} \left(e^{j\Theta_0 t} + e^{-j\Theta_0 t} \right)
\]
What integer index, \( k \), of the DFT, \( X[k] \), corresponds to the frequency closest to the complex component \( \frac{1}{2} e^{j\Theta_0 t} \) present in the input signal?
**Provide your answer as an integer value.**
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