**6.1 Short-time Fourier transform**

Short-time Fourier transformation (STFT) is used to analyze a nonstationary signal in the frequency-domain. The signal is sliced and subjected to Fourier transform. Segmenting the signal is called time domain windowing, and the time localized signal is defined by *St* (τ) <sup>=</sup> *<sup>S</sup>* (τ)*h*(<sup>τ</sup> <sup>−</sup> *<sup>t</sup>*), where *h*(*t*) is the window function centered at time t. The equation for STFT is given by (2).

$$\mathcal{S}\_t(\alpha, t) \;= \frac{1}{\sqrt{2n}} \int \mathcal{S}\_t(\tau) h(\tau - t) \, e^{-i\alpha t} \, d\tau \tag{2}$$

Mitchell et al. [60] used cross time-frequency analysis to diagnose hypertension of the GM muscle. The study included 57 elderly people with 10 younger adults. Reduced Interference distribution (RID) was utilized to remove cross terms implementing time smoothing window and frequency smoothing window. A Hanning frequency smoothing window was chosen. In the study of gait, it is necessary to consider a time-localized cross-correlation between two signals, such as left and right muscle groups responsible for gait [60]. Hence, cross Wigner distribution (CWD) was selected to preserve the phase information. The results revealed statistical significance for several time-frequency parameters of sEMG between control group and persons with neuropathy, diabetes, osteoporosis, and arthritis patients [60]. STFT does not adopt an optimal time window or frequency resolution for non-stationary signals [7]. For the implementation of FFT and STFT the signals are considered to be stationary [8]. The problem or resolution can be overcome by continuous wavelet transform (CWT) [8]. Multitaper analysis is another and perhaps more efficient method for power spectral analysis to deal with non-stationary signals [61, 62].

### **6.2 The wavelet transform**

Wavelet transform such as Multitaper is well suited for non-stationary signals. Wavelet transform elicits good localization of energy when the MUAP shape matches that of the wavelet [8]. Continuous wavelet transform (CWT) of bandpass filtered EMG showed alteration in the motor unit among stroke patients when a foot drop stimulator device was used (FDS) [63]. Energy localization below 100 Hz that resulted from foot drop was caused by slow motor unit recruitment. The neuromuscular activation improved with FDS. The time-frequency plot for Gastrocnemius showed that peak energy localization shifted from 50 to 100 Hz as a neuromuscular strategy [63]. Instantaneous mean frequency (IMNF) is the average frequency of power density spectrum of a signal and is computed from time-frequency distribution, W(*f, t*) [63], where W is obtained from continuous wavelet transformation defined by (3) and (4).

$$\text{IMNF}(t) \quad = \frac{\sum\_{j=1}^{N} f\_i W(f\_i, t)}{\sum\_{j=1}^{N} W(f\_i, t)} \tag{3}$$

$$\mathcal{W}\{\mathcal{X},\mathcal{Y}\}\quad = \frac{1}{\sqrt{\mathcal{X}}} \int\_{-\phi}^{\omega} \mathcal{Y}(t)\Psi. \frac{\langle t-\mathcal{Y} \rangle}{\mathcal{X}} dt \tag{4}$$

In the above, *x* is the scaling factor that controls the width of the wavelet, *y* controls its location in time, ψ is the mother wavelet function and *y*(*t*) is the signal. Instantaneous mean frequency can also be computed from the scalogram of CWT by its dimensional reduction. The scalogram has three dimensional space with time (x axis), frequency (y axis) and power (z axis) [63, 64]. In growing children, the higher IMNF level computed from scalogram revealed difference with respect to the children with cerebral palsy. The IMNF frequency component, unlike healthy children, decreased with age and maturation for children with cerebral palsy. IMNF also provided significant differences between the affected and unaffected site among stroke patients [63].
