DSP lab 4 part 1 done

6th-Semester-Spring-2024/DSP/Labs/Lab-04/lab-4.tex

@@ -10,7 +10,7 @@
 \usepackage[margin=1in]{geometry}
 
 
-\title{}
+\title{Discrete Fourier Transforms and Z-Transforms}
 
 \author{Aidan Sharpe \& Elise Heim}
 
@@ -26,16 +26,43 @@
 \subsection{The Discrete Fourier Transform (DFT)}
 Given a signal, $x[n]$, it's $N$-point DFT is given by
 \begin{equation}
-	X_k = \sum_{n=0}^{N-1} x[n] W_N^{kn}
+	X_k = \sum_{n=0}^{N-1} x[n] W_N^{kn},
 	\label{eqn:DFT_def}
 \end{equation}
 where $W_N = e^{-j2\pi/N}$. The discrete Fourier transform is the sampled version of the discrete time Fourier transform (DTFT), which is a continuous function. More specifically, the $N$-point DFT contains $N$ samples from the continuous DTFT.
-
-For example, consider the signal $x[n] = (-1)^n$ for $0 \le n \le N-1$. By evaluating the sum shown in equation \ref{eqn:DFT_def}, the $N$-point DFT of $x[n]$ is found to be
+\\
+\\
+For example, consider the signal $x[n] = (-1)^n$ for $0 \le n \le N-1$. By evaluating the sum shown in equation \ref{eqn:DFT_def} as a truncated geometric series, the $N$-point DFT of $x[n]$ can be found. All truncated geometric series are evaluated as
 \begin{equation}
-	X_k = {1 + e^{-j2\pi k} \over 1 + e^{-j2\pi k / N}}.
-	\label{eqn:DFT_ex}
+    \sum_{k=0}^{n-1} a r^k = 
+    \begin{cases}
+        an & r = 1 \\
+        a\left({1 - r^n \over 1 - r}\right) & r \ne 1
+    \end{cases},
 \end{equation}
+where $r$ is the common ratio between adjacent terms. For the $N$-point DFT of $x[n]$, the common ratio is $-W_N^k$, which takes a value of 1 for $k = {N\over2}$. Therefore, the $N$-point DFT of $x[n]$ is
+\begin{equation}
+    X[k] = \begin{cases}
+        N & k = {N \over 2} \\
+        \left({1 - (-W_N^k)^N \over 1 - (-W_N^k)}\right) & k \ne {N \over 2}
+    \end{cases}.
+    \label{eqn:DFT_N_point}
+\end{equation}
+The $N$-point DFT of $x[n]$, where $N=8$ is seen in figure \ref{fig:N_point_DFT}. It only has a non-zero value for $k={N\over2}=4$. This is the case for all even-number-point DFTs. Therefore only odd-number-point DFTs should be used.
+\begin{figure}[h]
+    \center
+    \includegraphics[width=0.5\textwidth]{N8_point_DFT.png}
+    \caption{The $N$-point DFT of $x[n]$, where $N=8$}
+    \label{fig:N_point_DFT}
+\end{figure}
+For example, the 9-point DFT of $x[n]$, where $N=8$ is seen in figure \ref{fig:9_point_DFT}. While  equation \ref{eqn:DFT_N_point} cannot be used because there are a different number of samples for the DFT and the input signal, the overall DFT is more useful than the 8-point DFT.
+\begin{figure}[h]
+        \center
+    \includegraphics[width=0.5\textwidth]{Q9_point_DFT.png}
+    \label{fig:9_point_DFT}
+\end{figure}
+
+
 
 \subsection{The Z-Transform}
 Given a discrete signal, $x[n]$, its z-transform is given by