By applying a form of JTFA, you can identify not only the frequency content of a signal, but also how that content evolves over time. Figure 3 shows clearly different representations for the low-to-high and high-to-low frequency chirp signals.
Figure 3. The JTFA of low-to-high (left) and high-to-low (right) frequency chirp signals are clearly different and showing both frequency content (y-axis) and time evolution (x-axis).
JTFA is a set of transforms that maps a one-dimensional time domain signal into a two-dimensional representation of energy versus time and frequency. In the examples above, JTFA shows the frequency content of a signal and the change in frequency with time. There are numerous applications in both research and industry for JTFA. Examples include speech analysis, telecommunications, underwater acoustics, bioacoustics, geophysics, and structural analysis.
There are a number of different transforms available for JTFA. Each transform type shows a different time-frequency representation. The Short Time Fourier Transform (STFT) is the simplest JTFA transform (and the easiest to compute). For the STFT, you apply the well-known Fast Fourier Transform (FFT) repeatedly to short segments of a signal at ever-later positions in time. You can display the result on a 3-D graph or a so-called 2-D 1/2 representation (the energy is mapped to light intensity or color values).
The STFT technique suffers from an inherent coupling between time resolution and frequency resolution (increasing the first decreases the second, and vice versa). This coupling can skew the measurements that you can derive from the transform, such as average instantaneous frequency.
Other JTFA methods and transforms can yield a more precise estimate of the energy in a given Frequency-Time domain. Some options include:
- Gabor spectrogram
- Wavelet transform
- Wigner distribution
- Cohen class transforms