Continuous Fourier Transform (CFT)
The CFT is defined for continuous-time signals, which are basically a signals that can take on any value at any time.
The continuous Fourier transform (CFT) of a signal x(t) can be defined as follows:
X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt
where f is the frequency in Hertz.
Notation that used in the CFT formula is:
- x(t) is the time-domain signal.
- X(f) is the frequency-domain signal.
- j is the imaginary unit.
- e −j2πft is the complex exponential function.
Derivation of the CFT
The CFT can be easily derived from the Fourier series of an periodic signal. The Fourier series of a periodic signal x(t) with period T is given by:
x(t) = \sum_{n=-\infty}^{\infty} c_n e^{j2\pi n\frac{t}{T}}
Here Cn are the Fourier coefficients of the signal.
The CFT can be obtained by simply taking the limit of the Fourier series as the period T approaches to the infinity. In this limit, the Fourier coefficients become a continuous functions of frequency, and the Fourier series becomes the CFT.
Fourier Transform in Circuit Analysis
In this article, we will study about the Fourier transform analysis or Fourier Transform in Circuit Analysis. The Fourier transform is basically a mathematical operation that decomposes a signal into its constituent frequency components. In simple words, it converts a signal from the time domain to the frequency domain. The time domain will represent the signal as a function of time, while the frequency domain represents the signal as a function of frequency.