Short-Time Fourier Transform Advanced Signal Processing Toolkit
The signal is expressed as the sum of sinusoids via the discrete Fourier transform. The discrete Fourier transform of the signal reflects the spectrum content of a signal across time, but it does not reveal the exact time at which the spectral content occurs because the sinusoids have an indefinite time length. Nonetheless, analyzing a signal's spectral content over a brief time span can be helpful in certain circumstances. The STFT is a useful tool for spectral content evaluation on short time intervals.
The time-domain input signal is divided into many disconnected or overlapped blocks using the STFT, also known as the windowed Fourier transform or the sliding Fourier transform. Each block is then subjected to the discrete Fourier transform. Functions with window functions, sometimes known as sliding windows, have amplitudes that taper smoothly and gradually toward zero at the margins. The generated STFT shows the spectral content of the signal at each corresponding time period since each block spans a separate time period. You may see the signal's spectral content over various time intervals by adjusting the sliding window. As a result, the STFT is a function of frequency and time that shows how a signal's spectral content changes over time. The output of windowed Fourier transformations is stored in a complex-valued, two-dimensional array known as the STFT coefficients. A phase time-frequency spectrum is formed by the STFT coefficients' phases, and a magnitude time-frequency spectrum is formed by their magnitudes.
One of the easiest methods for doing time-frequency analysis is the STFT, which can also make the notion of time-frequency analysis simple to comprehend. Due to its use of the fast Fourier transform (FFT), the STFT is computationally efficient.
But unless you take extra care, the STFT is not invertible, which means you can't use the STFT of a signal to reconstruct the time-domain waveform. For instance, you cannot reconstruct the signal in the time domain from the STFT if you step the sliding window without overlap. One type of invertible algorithm that is a particular instance of the STFT is the discrete Gabor transform. Discrete Gabor expansion is the name given to the inverse of the discrete Gabor transform.
When you don't need signal reconstruction or when you require the phase spectrum, you can apply the linear STFT approach. The phase spectrum, for instance, could be useful in applications involving computerized speech recognition. Utilize the quadratic STFT spectrogram approach, which is the square of the linear STFT, if you just need the magnitude spectrum for a given application. Due to its ease of use, the STFT spectrogram is among the most widely used quadratic time-frequency analysis techniques.
Window Type and Window Length
The product of the time resolution and the frequency resolution is typically used to describe the time-frequency resolution of the STFT. The time-frequency resolution of the STFT is dependent on the kind of window you choose. For time-frequency resolution, the Gaussian window is the best option.
The STFT's temporal resolution and frequency resolution are likewise impacted by the window length. Because narrow windows have a wide bandwidth and a short lifetime, they produce a coarse frequency resolution but a good time resolution. Because wide windows have a long time length but a restricted frequency bandwidth, they produce a fine frequency resolution but a coarse time resolution. We refer to this phenomenon as the window effect. The STFT cannot be used to concurrently get a fine time resolution and a fine frequency resolution. The time resolution and frequency resolution of the STFT are constant throughout the time-frequency plane when it has a time-invariant window.
Use the wavelet or adaptive transform in the LabVIEW Wavelet Analysis Tools if you require an adaptable temporal and frequency resolution.
The properties of the signal you wish to investigate will determine the optimal window length. In order for the windowed signal block to be almost steady throughout the window interval, the window length must be both small enough and large enough that the windowed signal block's Fourier transform yields a respectable frequency resolution. Increase the window length if the signal's spectral content changes gradually over time and fine time resolution is not needed. Reduce the window length if the signal's spectral content shifts quickly and demands a precise time resolution. For instance, a time-domain window length of 25 ms is frequently used in voice signal processing.
If overlap exists, it is determined by the sliding window's step size. Overlap occurs when the step size is less than the window length. There is no overlap if the step size is larger than the window length. The STFT is more smoothly behaved along the time axis when the sliding window overlaps. Overlap, however, calls for extra memory and processing time. Overlapping the sliding window is not necessary if the spectral content evolves slowly and the signal length is big. To provide a smoother STFT, overlap the sliding window if the signal length is minimal.
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