Discrete Fourier transforms (DFT)
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| Discrete Fourier Transform Pairs and Properties (info) | |
|---|---|
| Definition Discrete Fourier Transform and its Inverse | |
| Let x[n] be a periodic DT signal, with period N. | |
| N-point Discrete Fourier Transform | $ X [k] = \sum_{n=0}^{N-1} x[n]e^{-j 2\pi \frac{k n}{N}} \, $ |
| Inverse Discrete Fourier Transform | $ \,x [n] = (1/N) \sum_{k=0}^{N-1} X[k] e^{j 2\pi\frac{kn}{N}} \, $ |
| Discrete Fourier Transform Pairs (info) | |||
|---|---|---|---|
| $ x[n] \ \text{ (period } N) $ | $ \longrightarrow $ | $ X_N[k] \ \ (N \text{ point DFT)} $ | |
| $ \ \sum_{k=-\infty}^\infty \delta[n+Nk] = \left\{ \begin{array}{ll} 1, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right. $ | $ \ 1 \text{ (period } N) $ | ||
| $ \ 1 \text{ (period } N) $ | $ \ N\sum_{m=-\infty}^\infty \delta[k+Nm] = \left\{ \begin{array}{ll} N, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right. $ | ||
| $ \ e^{j2\pi k_0 n} $ | $ \ N\delta[((k - k_0))_N] $ | ||
| $ \ \cos(\frac{2\pi}{N}k_0n) $ | $ \ \frac{N}{2}(\delta[((k - k_0))_N] + \delta[((k+k_0))_N]) $ | ||
| Discrete Fourier Transform Properties | |||
|---|---|---|---|
| $ x[n] \ $ | $ \longrightarrow $ | $ X[k] \ $ | |
| Linearity | $ ax[n]+by[n] \ $ | $ aX[k]+bY[k] \ $ | |
| Circular Shift | $ x[((n-m))_N] \ $ | $ X[k]e^{(-j\frac{2 \pi}{N}km)} \ $ | |
| Duality | $ X[n] \ $ | $ NX[((-k))_N] \ $ | |
| Multiplication | $ x[n]y[n] \ $ | $ \frac{1}{N} X[k]\circledast Y[k], \ \circledast \text{ denotes the circular convolution} $ | |
| Convolution | $ x(t) \circledast y(t) \ $ | $ X[k]Y[k] \ $ | |
| $ \ x^*[n] $ | $ \ X^*[((-k))_N] $ | ||
| $ \ x^*[((-n))_N] $ | $ \ X^*[k] $ | ||
| $ \ \Re\{x[n]\} $ | $ \ X_{ep}[k] = \frac{1}{2}\{X[((k))_N] + X^*[((-k))_N]\} $ | ||
| $ \ j\Im\{x[n]\} $ | $ \ X_{op}[k] = \frac{1}{2}\{X[((k))_N] - X^*[((-k))_N]\} $ | ||
| $ \ x_{ep}[n] = \frac{1}{2}\{x[n] + x^*[((-n))_N]\} $ | $ \ \Re\{X[k]\} $ | ||
| $ \ x_{op}[n] = \frac{1}{2}\{x[n] - x^*[((-n))_N]\} $ | $ \ j\Im\{X[k]\} $ | ||
| Other Discrete Fourier Transform Properties | |
|---|---|
| Parseval's Theorem | $ \sum_{n=0}^{N-1}|x[n]|^2 = \frac{1}{N} \sum_{k=0}^{N-1}|X[k]|^2 $ |
Go to Relevant Course Page: ECE 438
