Spectrum analysis using the discrete fourier transform

Discrete-time fourier transform

The Discrete-Time Fourier Transform (DTFT) is the primary theoretical tool for understanding the frequency content of a discrete-time (sampled) signal.The DTFT is defined as

The DTFT is very useful for theory and analysis, but is not practical for numerically computing a spectrum digitally, because

• infinite time samples means
  • infinite computation
  • infinite delay
• The transform is continuous in the discrete-time frequency, ω

For practical computation of the frequency content of real-world signals, the Discrete Fourier Transform (DFT) is used.

Discrete fourier transform

The DFT transforms $N$ samples of a discrete-time signal to the same number of discrete frequency samples, and is defined as

Relationships between dft and dtft

Dft and discrete fourier series

The DFT gives the discrete-time Fourierseries coefficients of a periodic sequence ( $x(n)=x(n+N)$ ) of period $N$ samples, or

The DFT can thus be used to exactly compute the relative values of the $N$ line spectral components of the DTFT of any periodic discrete-time sequence with an integer-length period.

Dft and dtft of finite-length data

When a discrete-time sequence happens to equal zero for all samples except for those between $0$ and $N-1$ , the infinite sum in the DTFT equation becomes the same as the finite sum from $0$ to $N-1$ in the DFT equation. By matching the arguments in the exponential terms, we observe that theDFT values exactly equal the DTFT for specific DTFT frequencies ${\omega }_k=\frac{2\pi k}{N}$ . That is, the DFT computes exact samples of the DTFT at $N$ equally spaced frequencies ${\omega }_k=\frac{2\pi k}{N}$ , or

$$X({\omega }_k=\frac{2\pi k}{N})=\sum_{n=-\infty }^{\infty } x(n)e^{-(i{\omega }_{k}n)}=\sum_{n=0}^{N-1} x(n)e^{-\left(\frac{i\times 2\pi nk}{N}\right)}=X(k)$$

Dft as a dtft approximation

In most cases, the signal is neither exactly periodic nor truly of finite length; in such cases, the DFT of a finite block of $N$ consecutive discrete-time samples does not exactly equal samples of the DTFT at specific frequencies. Instead, the DFT gives frequency samples of a windowed (truncated) DTFT $$\hat{X}({\omega }_k=\frac{2\pi k}{N})=\sum_{n=0}^{N-1} x(n)e^{-(i{\omega }_{k}n)}=\sum_{n=-\infty }^{\infty } x(n)w(n)e^{-(i{\omega }_{k}n)}=X(k)$$ where $w(n)=\begin{cases}1 & \text{if $0\le n< N$}\\ 0 & \text{if $\text{else}$}\end{cases}$ Once again, $X(k)$ exactly equals $X({\omega }_k)$ a DTFT frequency sample only when $\forall n, n\notin \left[0 , N-1\right]\colon x(n)=0$