Intepreting a two-sided spectrum.

• Kruum
In summary, in a two-sided spectrum, a complex component of 1-j2 at frequency f = -1000Hz would also have a magnitude of 1-j2 at frequency f = 1000Hz. The signal in the time domain would be a cosine wave with a frequency of 2000Hz and a phase shift of 180 degrees, and the amplitude would depend on the magnitude of the complex component at 1000Hz. To calculate the inverse of the coefficient c_n of the Fourier series, more information about the signal in the frequency domain would be needed.
Kruum

Homework Statement

In a two-sided spectrum, there's a complex component of 1-j2 at frequency f = -1000Hz. What is the component at 1000Hz and the signal in time plane?

The Attempt at a Solution

So the component at 1000Hz is 1-j2 as well. And the signal in time plane is $a*cos(2000\pi t)$, but what is the amplitude. Is it simply 2*(1-j2)? Is there a possibility to calculate an inverse of the coefficient c_n of the Fourier series?

I would like to address the questions in the forum post and provide some additional information.

Firstly, in a two-sided spectrum, the complex component at frequency f = -1000Hz is represented by a negative frequency, which means it has a phase shift of 180 degrees. Therefore, the component at 1000Hz would also have a phase shift of 180 degrees, but the magnitude would remain the same. So, the component at 1000Hz would also be 1-j2.

Secondly, the signal in the time domain can be represented as a cosine wave with a frequency of 2000Hz and a phase shift of 180 degrees. The amplitude of the cosine wave would depend on the magnitude of the complex component at 1000Hz. If we assume that the amplitude of the cosine wave is 1, then the amplitude of the complex component at 1000Hz would be 1/2. This can be calculated using the formula for the magnitude of a complex number: |z| = sqrt(a^2 + b^2), where a and b are the real and imaginary parts of the complex number respectively.

Lastly, to calculate the inverse of the coefficient c_n of the Fourier series, we would need more information about the signal in the frequency domain. The Fourier series coefficients are calculated using the formula c_n = (1/T) * integral of f(t)*e^(-i2πnft)dt, where T is the period of the signal and f(t) is the signal in the time domain. So, without knowing the specific signal in the frequency domain, it would not be possible to calculate the inverse of the coefficient c_n.

I hope this helps to clarify some of the questions in the forum post. Please let me know if you have any further questions.

The component at 1000Hz would also be 1-j2, as the spectrum is symmetrical around 0Hz. As for the signal in the time plane, it would be a cosine wave with a frequency of 2000Hz, but the amplitude cannot be determined without more information. The Fourier series coefficient c_n would not be useful in this case, as the signal is not periodic. It would be necessary to know the specific values of a and the phase of the cosine wave to calculate the amplitude.

1. What is a two-sided spectrum?

A two-sided spectrum is a type of frequency analysis that displays the amplitude and phase of a signal across a range of frequencies. It is often used in signal processing and can provide valuable information about the frequency components of a signal.

2. How is a two-sided spectrum different from a one-sided spectrum?

A one-sided spectrum only displays the positive frequencies of a signal, while a two-sided spectrum displays both positive and negative frequencies. This can be useful when analyzing signals with both positive and negative frequency components.

3. What is the significance of the amplitude and phase in a two-sided spectrum?

The amplitude represents the strength or magnitude of a frequency component in the signal, while the phase represents the timing or synchronization of that component with the rest of the signal. These values can provide insights into the characteristics and behavior of the signal.

4. How is a two-sided spectrum typically visualized?

A two-sided spectrum is often visualized using a graph, with the horizontal axis representing frequency and the vertical axis representing amplitude or phase. The resulting plot is typically symmetrical, with the negative frequencies mirrored on the other side of the y-axis.

5. What are some common applications of interpreting a two-sided spectrum?

A two-sided spectrum can be used in a variety of fields, including audio and image processing, telecommunications, and medical imaging. It can help identify specific frequency components in a signal, detect anomalies or patterns, and aid in the design and optimization of systems and devices.

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