Syncing to work on interview with entrepreneur essay

6th-Semester-Spring-2024/DSP/Notes/dsp-notes.tex

@@ -734,4 +734,39 @@ $$h[n] = -7(0.9)^n u[-n-1] - 6(0.8)^n u[n]$$
 {
 	For FIR systems, if the Z-transform does not converge at $|z|= 0$ or $|z| = \infty$, they are not considered poles, because only IIR systems can have poles.
 }
+
+\chapter{The Discrete Fourier Transform}
+The discrete Fourier transform is the sampled DTFT. Since the DTFT is a continuous function, it cannot be analyzed in the same way as a discrete signal. The N-point DFT is defined as:
+$$X[k] = \sum_{n=0}^{N-1} x[n] W_N^{kn}$$
+where $W_N^k = e^{-j2\pi k / N}$.
+
+\ex{}
+{
+	Find the N-point DFT of $x[n] = u[n] - u[n-N]$.
+	$$\sum_{n=0}^{N-1} s^n = \begin{cases}
+		{1 - s^N \over 1 - s} & s \ne 1 \\
+		N & s = 1
+	\end{cases}$$
+	
+	When $k = 0$:
+	$$X[0] = \sum_{n=0}^{N-1} (1)^n = N$$
+	When $k \ne 0$:
+	$$X[k] = {1 - W_N^{kN} \over 1 - W_N^k} = 0$$
+
+	These points are zero because they correspond to the zero-crossings of the $\text{sinc}$ function, which is the Fourier transform of a unit pulse.
+}
+
+\section{The Inverse DFT}
+\ex{}
+{
+	The 10-point DFT of $x[n]$ is given as:
+	$$X[k] = 2\delta[k] + 1, k \in [0,9] \cap \mathbb{Z}$$
+	Find the 10-point signal $x[n]$
+}
+
+\chapter{FIR Filter Design}
+Consider $h[n]$ with finite support $0 \le n \le M$. The filter is said to have an order $M$ with $M+1$ taps or coefficients. FIR systems are always BIBO stable. Focus will be on Type-1 FIR filter, where $M$ is even, so the number of taps is odd, and $h[n]$ is symmetric over the center coefficient $h\lt[M\over2\rt]$.
+
+
+
 \end{document}