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The Continuous Time Fourier Transform (CTFT)
The CTFT transforms an infinite length continuous signal in the time domain into an infinite length signal in the frequency domain. According to Wikipedia's definition (I couldn't find a better one, unfortunately), the Fourier transform is an operation that transforms one complex-valued function of a real variable into another. In signal processing, the domain of the original function is typically in the time domain, while the domain of the new function is typically called the frequency domain. The new function itself is called the frequency domain representation of the original function and it describes which frequencies are present in the original function.
The formulae are:-
CTFT:
$ X(\omega) = \int_{-\infty}^{\infty} \! x(t)e^{-j \omega t} dt $
Inverse CTFT:
$ x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \! X(\omega)e^{j \omega t} dw $
Example:
Let $ x(t) = \delta (t) $
$ \begin{align} X(\omega) &= \int_{-\infty}^{\infty} \! x(t)e^{-j \omega t} dt \\ &= \int_{-\infty}^{\infty} \! \delta (t)e^{-j \omega t} dt \\ &= 1\end{align} $
Therefore, CTFT of $ \delta (t) = 1 $
Another good example to look at is this one
CTFT Properties
Linearity:
$ CTFT[x(t)] = X(\omega) $
Then $ CTFT[\alpha x_1(t) + \beta x_2(t)] = \alpha X_1(\omega) + \beta X_2(\omega) $
Time Reversal:
$ CTFT[x(t)] = X(\omega) $
Then $ CTFT[x(-t)] = X(-\omega) $
Time/Frequency Shift:
Time: $ CTFT[x(t-T)] = X(\omega)e^{-j \omega T} $
Frequency: $ CTFT[x(t)e^{j W t}] = X(\omega - W) $
Time Scaling:
$ CTFT[x(\alpha t)] = \frac{1}{|\alpha |} \cdot X(\frac{\omega}{\alpha}) $
Conjugate Symmetry:
Assume x(t) is real,
$ CTFT[x(t)] = X(\omega) $
Then $ X(\omega) = X^*(\omega) $
Time Domain Multiplication/Convolution:
$ CTFT[x(t)\cdot y(t)] = \frac{1}{2\pi}X(\omega)*Y(w) $
$ CTFT[x(t)*y(t)] = X(\omega)Y(\omega) $
The Discrete Time Fourier Transform (DTFT)
(Wikipedia) The DTFT is one of the specific forms of Fourier analysis. It transforms a function in the time domain into the frequency domain, just like the CTFT, but requires the input function to be discrete. To obtain a discrete input function, the input function is created by sampling a continuous function.
The DTFT frequency-domain representation is always a periodic function. Since one period of the function contains all of the unique information, it is sometimes convenient to say that the DTFT is a transform to a "finite" frequency-domain (the length of one period), rather than to the entire real line.
DFT:
$ X(\omega) = \sum_{n=-\infty}^{\infty} x[n] \,e^{-j \omega n} $
IDFT:
$ x[n] = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega)e^{j \omega n} d\omega $
Facts:
$ X(\omega) $ is periodic with period $ 2\pi $
Discrete in time, but continuous in frequency
Example:
$ Let x_n = a^n u[n] $ |a|< 1
$ \begin{align} X(\omega) &= \sum_{-\infty}^{\infty} \! a^n u[n]e^{-j \omega n} \\ &= \sum_{0}^{\infty} \! (ae^{-j \omega})^n \\ &= \frac{1}{1 - ae^{-j \omega}}\end{align} $
DTFT Properties
Linearity:
$ DTFT[x(n)] = X(\omega) $
Then $ DTFT[\alpha x_1(n) + \beta x_2(n)] = \alpha X_1(\omega) + \beta X_2(\omega) $
Time Reversal:
$ DTFT[x(n)] = X(\omega) $
Then $ DTFT[x(-n)] = X(-\omega) $
Time/Frequency Shift:
Time: $ DTFT[x(n-T)] = X(\omega)e^{-j \omega T} $
Frequency: $ DTFT[x(n)e^{j \omega_0 n}] = X(\omega - \omega_0) $
Conjugate Symmetry:
Assume x(t) is real,
$ DTFT[x(n)] = X(\omega) $
Then $ X(\omega) = X^*(-\omega) $
Time Domain Multiplication/Convolution:
$ DTFT[x*y(n)] = X(\omega)Y(\omega) $
Modulation
$ DTFT[x(n)h(n)] = \frac{1}{2\pi} \ X(\omega) * H(\omega) $
where $ * $ in this case represents the circular convolution.
$ X(\omega) * H(\omega) = \int_{0}^{2\pi}X(\gamma)H(\omega - \gamma) d\gamma $
Parseval's Relation
$ \sum_{n = -\infty}^{\infty}|x(n)|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi}|X(\omega)|^2 d\omega $
