Discrete Fourier transform (DFT)
The DFT is defined for discrete-time signals, which are a signals that can only take on certain values at specific certain times.
The discrete Fourier transform (DFT) of a discrete-time signal x[n] can be defined as follows:
X[k] = \sum_{n=0}^{N-1} x[n] e^{-j2\pi kn/N}
Here k is the frequency index and N is the length of the particular signal signal.
Notation that used in the DFT formula is:
- x[n] is the discrete-time signal.
- X[k] is the frequency-domain signal.
- j is the imaginary unit.
- e −j2πkn/N
- is the complex exponential function.
Derivation of the DFT
In simple terms CFT is basically defined for continuous-time signals, while the DFT is defined for discrete-time signals. The DFT is mostly used the type of Fourier transform in circuit analysis, as most electronic circuits which operate on discrete-time signals.
The DFT of a discrete-time signal x[n] is denoted by the X[k] and is defined as follows:
X[k] = \sum_{n=0}^{N-1} x[n] e^{-j2\pi kn/N}
Here k is the frequency index and the N is the length of the signal.
The DFT can be derived from the CFT by simply sampling the CFT at discrete frequencies:
X[k] = X(f = k/N)
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.