**6. Frequency and time-frequency analysis**

EMG power spectrum estimation methods can be categorized into parametric and nonparametric techniques. The spectral methods include fast Fourier transform (FFT), multitaper analysis and short-time Fourier transform (STFT) and wavelet transform. The difference between FFT and Wavelet Transformation is that FFT is localized to the frequency domain whereas the latter is localized to time-frequency analysis. Hu [53] recorded cortical and spinal somatosensory evoked potential (CSEP and SSEP), cortical motor evoked potential (CMEP) and spinal cord evoked potential (SCEP). The short time Fourier transformation was applied to the CSEP signal with a Hanning window [53]. The results revealed that the time-frequency analysis is a better marker for spinal injury than time domain analysis. The peak power after spinal injury had lesser energy with more dispersion in time-frequency scale.

The EMG time series signal can be analyzed in the frequency domain for the diagnosis of gait disorders. The frequency spectrum for EMG signals is in range of 0–500 Hz [54]. The FFT algorithm [55] computes the discrete Fourier transform (DFT) of EMG signal more efficiently. The FFT decomposes the EMG signals into periodic sine and cosine waves. We computed the FFT of EMG signal recorded from the Vastus Medialis (VM) during walking (**Figure 4**).

#### **Figure 4.**

*(A) sEMG signal from VM during walking in time domain; (B) frequency domain representation of the signal using FFT.*

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signals [61, 62].

*A Review of EMG Techniques for Detection of Gait Disorders*

The FFT allows computation of power spectra by squaring of FFT's magnitude [56]. In Parkinson disease, the spectral power of the signal has lower amplitude for the usual tremor than for the unusual tremor, which has peak amplitude of 4–6 Hz during an atypical tremor [15]. The signals associated with nonperiodic tremors are differentiable with FFT [57]. The EMG signal from neuropathic patients with SCI also exhibited distinct power spectrum density and amplitude in comparison to healthy individuals [58]. The application of FFT to the EMG envelope revealed muscle burst discharge in frequency domain ranging from 4 to 7 Hz [15]. Average power spectra computed from fractionated EMG of ALS mice by FFT was significantly higher than the control group. In the ALS group the spectra were skewed towards higher frequency content but single unit recordings revealed the absence of higher motor neuron (MN) frequencies or shortening of MN frequency in ALS mice [32], due to small type firing neurons improperly increasing firing frequency. This phenomenon results in co-contraction thus producing fractionated EMG. Co-contraction in muscles can also be observed in spinal cord injured patients [32]. In a study, EMG signals from lower limbs of dystonic and nondystonic participants while walking were recorded. The non-dystonic participants were also patients suffering from other gait disorders. The power spectral density was computed using FFT with the Welch method of 50% overlap. The median power frequency (MdPF) and total power in low frequency were calculated for each muscle. The results revealed that MdPF for dystonic muscles had shifted to low frequencies and a concurrent increase in total power percentage in low-frequency range was observed [59]. Thus, frequency analysis of EMG signal not only provides us with distinction between normal and abnormal gait behavior but also specific

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

> √ \_\_\_ 2*n*

(τ) <sup>=</sup> *<sup>S</sup>* (τ)*h*(<sup>τ</sup> <sup>−</sup> *<sup>t</sup>*), where *h*(*t*) is the window function

. *d* (2)

∫*S* (τ)*h*(τ − *t*) *e*<sup>−</sup>*i<sup>t</sup>*

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

*DOI: http://dx.doi.org/10.5772/intechopen.84403*

gait abnormalities can be distinguished.

*St*(ω,*t*) = \_\_\_1

centered at time t. The equation for STFT is given by (2).

**6.1 Short-time Fourier transform**

localized signal is defined by *St*
