Collective Table of Formulas

Discrete-time (DT) Fourier Transforms Pairs and Properties

(used in ECE301, ECE438, ECE538)



DT Fourier transform and its Inverse
DT Fourier Transform $ \,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\, $
Inverse DT Fourier Transform $ \,x[n]=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n} d \omega\, $
DT Fourier Transform Pairs
$ x[n] \ $ $ \longrightarrow $ $ \mathcal{X}(\omega) \ $
DTFT of a complex exponential $ e^{jw_0n} \ $ $ \ 2\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ $
(info) DTFT of a rectangular window $ w[n]= \ $ $ \text{add formula here} \ $
$ a^{n} u[n], |a|<1 \ $ $ \frac{1}{1-ae^{-j\omega}} \ $
$ (n+1)a^{n} u[n], |a|<1 \ $ $ \frac{1}{(1-ae^{-j\omega})^2} \ $
$ \sin\left(\omega _0 n\right) u[n] \ $ $ \frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right) $
$ \cos\left(\omega _0 n\right) \ $ $ \pi \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)+\delta(\omega+\omega_0-2\pi k)) $
$ \sin\left(\omega _0 n\right) \ $ $ \frac{\pi}{j} \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)-\delta(\omega+\omega_0-2\pi k)) $
$ 1 \ $ $ 2\pi\sum^{\infty}_{k=-\infty}\delta(\omega-2\pi k) $
DTFT of a Periodic Square Wave

$ \left\{\begin{array}{ll}1, & |n|<N_1,\\ 0, & N_1<|n|\leq\frac{N}{2}\end{array} \right. \text{ and } x[n+N]=x[n] $

$ 2\pi\sum^{\infty}_{k=-\infty}a_k\delta(\omega-\frac{2\pi k}{N}) $
$ \sum^{\infty}_{k=-\infty}\delta[n-kN] $ $ \frac{2\pi}{N}\sum^{\infty}_{k=-\infty}\delta(\omega -\frac{2\pi k}{N}) $
$ \delta [n] \ $ $ 1 \ $
$ u[n] \ $ $ \frac{1}{1-e^{-j\omega}}+\sum^{\infty}_{k=-\infty}\pi\delta(\omega-2\pi k) $
$ \delta[n - n_0] \ $ $ e^{-j\omega n_0} $
$ (n + 1)a^n u[n], \quad |a| < 1 $ $ \frac{1}{(1-ae^{-j\omega})^{2}} $
DT Fourier Transform Properties
$ x[n] \ $ $ \longrightarrow $ $ \mathcal{X}(\omega) \ $
multiplication property $ x[n]y[n] \ $ $ \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\theta)Y(\omega-\theta)d\theta $
convolution property $ x[n]*y[n] \ $ $ X(\omega)Y(\omega) \! $
time reversal $ \ x[-n] $ $ \ X(-\omega) $
Differentiation in frequency $ \ nx[n] $ $ \ j\frac{d}{d\omega}X(\omega) $
Linearity $ ax[n]+by[n] \ $ $ aX(\omega)+bY(\omega) \ $
Time Shifting $ x[n - n_0] \ $ $ e^{-j\omega n_0}X(\omega) $
Frequency Shifting $ e^{j\omega_0 n}x[n] $ $ X(\omega - \omega_0) \ $
Conjugation $ x^* [n] \ $ $ X^* (-\omega) \ $
Time Expansion $ x_{(k)}[n]=\left\{\begin{array}{ll}x[n/k], & \text{ if n = multiple of k},\\ 0, & \text{else.}\end{array} \right. $ $ X(k\omega) \ $
Differentiating in Time $ x[n] - x[n - 1] \ $ $ (1 - e^{-j\omega}) X (\omega) \ $
Accumulation $ \sum^{n}_{k=-\infty} x[k] $ $ \frac{1}{1-e^{-j\omega}}X(\omega) $
Symmetry $ x[n] \ \text{ real and even} \ $ $ X(\omega) \ \text{ real and even} \ $
$ x[n] \ \text{ real and odd} \ $ $ X(\omega) \ \text{ purely imaginary and odd} \ $
Other DT Fourier Transform Properties
Parseval's relation $ \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|X( \omega )|^2d\omega $




Back to Collective Table

Alumni Liaison

Have a piece of advice for Purdue students? Share it through Rhea!

Alumni Liaison