Power Spectral Density (PSD) of White Noise
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White noise is a random signal that has equal intensity at all frequencies. It is characterized by a flat power spectral density (PSD), meaning that the power of the noise is evenly distributed across all frequencies. In other words, the PSD of white noise is constant.

Ideal Frequency Detector
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An ideal frequency detector is a device that is able to accurately determine the frequency of an input signal. It operates by comparing the input signal to a reference frequency and producing an output signal that indicates the difference between the input and reference frequencies.

Effect of White Noise on Frequency Detector
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When white noise is present at the input of an ideal frequency detector, it can affect the accuracy and reliability of the detector's output. The white noise adds random fluctuations to the input signal, which can introduce errors in the frequency detection process.

Power Spectral Density of Output Noise
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The power spectral density of the noise at the output of the frequency detector can be different from that of the input noise. This is because the frequency detection process involves nonlinear operations, such as multiplication and filtering, which can alter the spectral characteristics of the noise.

In the case of an ideal frequency detector, the power spectral density of the output noise is parabolic. This means that the power of the noise is concentrated at the low and high-frequency ends of the spectrum, with lower power in the middle frequencies.

Reason for Parabolic PSD
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The parabolic nature of the output noise PSD is a result of the frequency detection process. When the input signal and reference frequency are multiplied together, new frequencies are generated as a result of the modulation process. These new frequencies are centered around the sum and difference of the input and reference frequencies.

As a result, the output noise PSD is spread over a wider frequency range compared to the input noise. The low and high-frequency components of the output noise correspond to the sum and difference frequencies, while the middle frequencies correspond to the original input and reference frequencies.

Conclusion
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In conclusion, the power spectral density of the noise at the output of an ideal frequency detector is parabolic. This is due to the nonlinear operations involved in the frequency detection process, which result in the generation of new frequencies and the redistribution of the noise power across the frequency spectrum.