Discrete-time (DT) Fourier Transforms Pairs and Properties
| 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 $ |
