Q1Signals and Systems

Let $x(n)$ and $y(n)$ be N-point sequences with N-point DFTs $X(k)$ and $Y(k)$.

1. Linearity: $ax(n)+by(n)\leftrightarrow aX(k)+bY(k)$. The DFT of a linear combination of signals is the linear combination of their DFTs.

2. Periodicity: $X(k+N)=X(k)$ and $x(n+N)=x(n)$. Both the time sequence and the frequency spectrum repeat with a period of N.

3. Circular Time Shift: If $x(n)$ is circularly shifted by $m$ samples to $x((n-m){)}_N$, its DFT becomes $X(k)e^{-j\frac{2\pi }{N}mk}$. A circular shift in time introduces a linear phase shift in frequency.

4. Circular Convolution: The circular convolution of two sequences in the time domain equates to the simple multiplication of their DFTs in the frequency domain. $x(n)⊛y(n)\leftrightarrow X(k)Y(k)$.

5. Complex Conjugation: If $x(n)$ is complex, the DFT of $x^{\ast }(n)$ is $X^{\ast }((-k){)}_N$. For real sequences, $X(k)=X^{\ast }(-k)$, meaning magnitude is even and phase is odd.

6. Parseval's Theorem: ${\sum }_{n=0}^{N-1}|x(n){|}^2=\frac{1}{N}{\sum }_{k=0}^{N-1}|X(k){|}^2$. The total energy in the discrete-time domain equals the total energy in the discrete-frequency domain.

A feedback control system (Closed-Loop System) is one in which the system's output is continuously measured and fed back to be compared with the desired input (reference). The difference between the two (the error signal) is used by a controller to adjust the system's input, effectively reducing the error and forcing the output to track the desired reference. They are highly robust against internal disturbances and parameter variations.