Editing Discrete Fourier transform (section)

=== Spectral analysis ===

When the DFT is used for [[signal spectral analysis]], the <math>\{x_n\}</math> sequence usually represents a finite set of uniformly spaced time-samples of some signal <math>x(t)\,</math>, where <math>t</math> represents time.  The conversion from continuous time to samples (discrete-time) changes the underlying [[continuous Fourier transform|Fourier transform]] of <math>x(t)</math> into a [[discrete-time Fourier transform]] (DTFT), which generally entails a type of distortion called [[aliasing]].  Choice of an appropriate sample-rate (see ''[[Nyquist rate]]'') is the key to minimizing that distortion.  Similarly, the conversion from a very long (or infinite) sequence to a manageable size entails a type of distortion called ''[[Spectral leakage|leakage]]'', which is manifested as a loss of detail (a.k.a. resolution) in the DTFT.  Choice of an appropriate sub-sequence length is the primary key to minimizing that effect.  When the available data (and time to process it) is more than the amount needed to attain the desired frequency resolution, a standard technique is to perform multiple DFTs, for example to create a [[spectrogram]].  If the desired result is a power spectrum and noise or randomness is present in the data, averaging the magnitude components of the multiple DFTs is a useful procedure to reduce the [[variance]] of the spectrum (also called a [[periodogram]] in this context); two examples of such techniques are the [[Welch method]] and the [[Bartlett method]]; the general subject of estimating the power spectrum of a noisy signal is called [[spectral estimation]].

A final source of distortion (or perhaps ''illusion'') is the DFT itself, because it is just a discrete sampling of the DTFT, which is a function of a continuous frequency domain.  That can be mitigated by increasing the resolution of the DFT.  That procedure is illustrated at {{slink|Discrete-time Fourier transform|Sampling the DTFT|nopage=y}}.
* The procedure is sometimes referred to as ''zero-padding'', which is a particular implementation used in conjunction with the [[fast Fourier transform]] (FFT) algorithm.  The inefficiency of performing multiplications and additions with zero-valued "samples" is more than offset by the inherent efficiency of the FFT.
* As already stated, leakage imposes a limit on the inherent resolution of the DTFT, so there is a practical limit to the benefit that can be obtained from a fine-grained DFT.
'''Steps to Perform Spectral Analysis of Audio Signal'''

'''1.Recording and Pre-Processing the Audio Signal'''

Begin by recording the audio signal, which could be a spoken password, [[music]], or any other [[sound]]. Once recorded, the audio signal is denoted as x[n], where n represents the discrete time index. To enhance the accuracy of spectral analysis, any unwanted noise should be reduced using appropriate [[Filter design|filtering techniques.]]

'''2.Plotting the Original Time-Domain Signal'''

After [[noise]] reduction, the audio signal is plotted in the time domain to visualize its characteristics over time. This helps in understanding the amplitude variations of the signal as a [[function of time]], which provides an initial insight into the signal's behavior.

'''3.Transforming the Signal from Time Domain to Frequency Domain'''

The next step is to transform the audio signal from the time domain to the frequency domain using the Discrete Fourier Transform (DFT). The DFT is defined as:

<math>X[k] = \sum_{n=0}^{N-1} x[n] \cdot e^{-j \frac{2 \pi}{N}kn}</math>

where N is the total number of samples, k represents the frequency index, and X[k] is the complex-valued frequency spectrum of the signal. The DFT allows for decomposing the signal into its constituent frequency components, providing a representation that indicates which frequencies are present and their respective magnitudes.

'''4.Plotting the Magnitude Spectrum'''

The magnitude of the frequency-domain representation X[k] is plotted to analyze the spectral content. The magnitude spectrum shows how the energy of the signal is distributed across different frequencies, which is useful for identifying prominent frequency components. It is calculated as:

<math>|X[k]| = \sqrt{\text{Re}(X[k])^2 + \text{Im}(X[k])^2}</math>