Discrete Time Fourier Transform (DTFT)

Sources

1. Lecture Notes
2. Lecture Pages (Module 8 Discrete-time Fourier Transform)

Check Your Understanding

Problem a.

Solve for the DTFT of the following signal:

$$x[n]=u[n]-u[n-6]$$

$$\begin{array}{rlrl}x[n]& =1,& & 0\le n\le 5\\ x[n]& =\{\underset{\uparrow }{1},1,1,1,1,1\}& & \\ X(\omega )& =\sum _{n=-\infty }^{\infty }x[n]e^{-j\omega n}& & \\ X(\omega )& =\sum _{n=0}^5(1)e^{-j\omega n}& & \\ X(\omega )& =1+e^{-j\omega }+e^{-j2\omega }+e^{-j3\omega }+e^{-j4\omega }+e^{-j5\omega }& & \end{array}$$

$a^n$

$$\sum _{n=0}^N{a}^n=\frac{1-a^{N+1}}{1-a}|a=e^{-j\omega }$$

$$\begin{array}{rl}X(\omega )& =\frac{1-(e^{-j\omega }{)}^6}{1-(e^{-j\omega })}\\ & =\boxed{\frac{1-e^{-j6\omega }}{1-e^{-j\omega }}}\end{array}$$

Problem b.

Solve for the DTFT of the following signal:

$$x[n]=2^{n}u[-n]$$

$$\begin{array}{rl}x[-n]& =2^{-n}u[n]\\ x[-n]& ={\frac{1}{2}}^{n}u[n]\\ X_1(\omega )& =\frac{1}{1-\frac{1}{2}{e}^{-j\omega }}\\ X(\omega )& =X_1(-\omega )\\ X(\omega )& =\boxed{\frac{1}{1-\frac{1}{2}{e}^{j\omega }}}\end{array}$$

Problem c.

Solve for the DTFT of the following signal:

$$x[n]={\left(\frac{1}{4}\right)}^n[n+4]$$

$$\begin{array}{rl}x[n]& ={\left(\frac{1}{4}\right)}^{-4}{\left(\frac{1}{4}\right)}^{n+4}u[n+4]\\ x[n]& =256{\left(\frac{1}{4}\right)}^{n+4}u[n+4]\\ x(z)& =256z^{4}Z\left\{{\left(\frac{1}{4}\right)}^{n}u[n]\right\}\\ & =\frac{256z^4}{1-\frac{1}{4}{z}^{-1}}\\ z& =e^{j\omega }\\ X(e^{j\omega })& =\boxed{\frac{256e^{j4\omega }}{1-\frac{1}{4}{e}^{-j\omega }}}\end{array}$$

Problem d.

Solve for the DTFT of the following signal:

$$\begin{array}{rlr}x[n]={\alpha }^n\sin ({\omega }_{0}n)u[n],& & |\alpha |<1\end{array}$$

$$\begin{array}{rl}X(z)& =\frac{\alpha z^{-1}\sin ({\omega }_0)}{1-2\alpha z^{-1}\cos {\omega }_0+{\alpha }^2{z}^{-2}}\\ z& =e^{j\omega }\\ X(\omega )& =\boxed{\frac{\alpha e^{-j\omega }\sin ({\omega }_0)}{1-2\alpha e^{-j\omega }\cos {\omega }_0+{\alpha }^2{e}^{-j2\omega }}}\end{array}$$

Problem e.

Solve for the DTFT of the following signal:

$$x[n]=\{-2,1,\underset{\uparrow }{0},1,2\}$$

$$\boxed{X(\omega )=-2e^{j2\omega }+e^{j\omega }+e^{-j\omega }+e^{-j2\omega }}$$

Example 10

Problem

Determine the magnitude spectrum and phase spectrum of the following signal:

$$\begin{array}{rlr}x[n]={\alpha }^{n}u[n],& & |\alpha |<1\end{array}$$

Find the DTFT and sketch both magnitude spectrum $|X(\omega )|$ and phase spectrum $\angle X(\omega )$

TODO

Example 11

Problem

$$\begin{array}{rlr}x[n]=a^{n}u[n],& & -1<\alpha <1\end{array}$$

• Find the DTFT
• Find the Energy Density Spectrum $S_{xx}(\omega )=|X(\omega ){|}^2$
• Sketch the Energy Density Spectrum
• Identify frequencies in which majority of the signal is concentrated

TODO