For Part A. You can solve this several different ways. If you don't know what to do, you should always resort to the definition of power. But in this case you should know that you can also find the power of a signal from its Fourier transform and this particular signal has a really simple Fourier transform. Normally I would just compute the power of the sinusoids from rms values but part A is leading into part B.beanus said:
The Continuous Time Fourier Transform (CTFT) is a mathematical tool used to analyze signals in the time domain and represent them in the frequency domain. It is particularly useful in studying Linear Time-Invariant (LTI) systems, which are systems that do not change over time. The CTFT converts a continuous time signal into a continuous frequency spectrum, providing valuable information about the frequency content of the signal.
The main difference between the CTFT and the DTFT is that the CTFT deals with continuous time signals, while the DTFT deals with discrete time signals. This means that the CTFT is used for signals that are continuous in both time and frequency domains, while the DTFT is used for signals that are discrete in both domains.
The Fourier Series is a special case of the CTFT, where the input signal is periodic. It is used to decompose a periodic signal into a series of sinusoidal components. The CTFT, on the other hand, is used for non-periodic signals and provides a continuous frequency spectrum.
The CTFT can be used to analyze LTI systems by converting the input signal into the frequency domain and then applying the system's frequency response to obtain the output signal in the frequency domain. The output signal can then be transformed back into the time domain using the inverse CTFT. This allows for a better understanding of how the system affects different frequencies in the input signal.
One of the main limitations of using the CTFT to analyze LTI systems is that it assumes the system is time-invariant, meaning that its behavior does not change over time. In real-world situations, many systems are not completely time-invariant, which can lead to errors in the analysis. Additionally, the CTFT is only applicable to signals that are absolutely integrable, which means that the signal's energy is finite over all time.