Mean absolute value of an EMG signal, in real time.

The signal and the target value.

The brief states the input is a sinewave, $x(t) = A\sin(\omega t)$. The mean absolute value is the time-average of $|x(t)|$ over a window $T$. For a sinewave with amplitude $A$ and a window that contains a whole number of half-cycles, the integral closes in form and the MAV settles to a constant.

For $x(t) = A\sin(\omega t)$, evaluating the integral gives $\mathrm{MAV} = \tfrac{2A}{\pi} \approx 0.637\,A$. The target value below is plotted as a dashed line so the chain output in later slides can be read against it.

Split the equation into two stages.

The MAV equation is the composition of two operators: an absolute value on the instantaneous signal, and an integral over a finite window. Each maps to a standard analog block.

A rectified sinewave is a sum of a DC term equal to $\tfrac{2A}{\pi}$ and a Fourier series of even harmonics of $\omega$. Averaging, or low-pass filtering with cutoff well below $2\omega$, isolates that DC term.

A precision rectifier implements $|x(t)|$.

A passive bridge of four diodes will rectify a sinewave, but each diode drop subtracts $V_D \approx 0.6$ V from the signal. EMG amplitudes are millivolts, so the diodes never turn on. A precision rectifier places the diode inside an op-amp feedback loop, dividing the diode drop by the open-loop gain so the output follows $|x(t)|$ down to nearly zero.

For a sinewave input $x(t) = A\sin(\omega t)$, the rectified output is

$|x(t)| \;=\; A\bigl|\sin(\omega t)\bigr| \;=\; \dfrac{2A}{\pi} \;-\; \dfrac{4A}{\pi}\!\sum_{k=1}^{\infty}\!\dfrac{\cos(2k\omega t)}{4k^{2}-1}$

The first term is the DC value the next stage must extract. The remaining terms are harmonics of $2\omega$, the lowest of which is at $2\omega$ itself. A low-pass filter with cutoff well below $2\omega$ removes them.

A low-pass filter extracts the mean.

Averaging in continuous time is integration over a window. A first-order RC low-pass filter is the analog approximation: the output is a weighted moving average of the input with an exponential window of time constant $\tau = RC$ and cutoff $f_c = 1/(2\pi\tau)$.

The rectified signal is a DC term at $\tfrac{2A}{\pi}$ plus harmonics at $2\omega, 4\omega, \dots$. A first-order low-pass filter at cutoff $f_c$ has magnitude $|H(f)| = 1/\sqrt{1 + (f/f_c)^2}$. Choosing $f_c \ll 2f_\mathrm{sig}$ attenuates every harmonic; the DC term passes unchanged, so the filter output settles to $\tfrac{2A}{\pi}$, which is the MAV.

The block diagram that computes MAV.

The two stages connect in series. The EMG sinewave enters the precision rectifier, the rectified signal feeds a first-order low-pass filter, and the filter output is the running mean absolute value. The chain is purely analog and runs continuously, so the output is available in real time without sampling or accumulation in software.

• Pre-amp (instrumentation amplifier) lifts the millivolt EMG to a level that drives the rectifier's op-amp safely above its input-referred noise.
• Precision rectifier outputs $|x(t)|$ with no diode-drop offset.
• Low-pass filter at cutoff $f_c \ll 2f_\mathrm{sig}$ extracts the DC component, which equals $\tfrac{2A}{\pi}$ and is the MAV.

MAV drives a prosthetic gripper.

A myoelectric prosthetic hand is controlled by surface EMG from the residual limb. The raw signal is bipolar and noisy; the controller cannot use it directly. The MAV envelope is the quantity that correlates with voluntary contraction force, so the chain from the previous slide is what the device runs in real time.

• Detection — a threshold on the MAV separates a voluntary contraction from baseline activity, so the gripper only engages when the user intends to grip.
• Proportional control — once above threshold, the MAV value sets motor drive, so a stronger contraction produces a firmer grip.
• Smoother choice — three options for the averaging stage are provided. The first-order LPF is the analog default; the moving-window average is the literal $\tfrac{1}{T}\!\int|x|\,\mathrm{d}t$; a 1-D Kalman filter adapts its gain as the variance evolves and gives the cleanest envelope.
• Latency vs. smoothness — the smoothing parameter $f_c$ sets the bandwidth of all three. Lower $f_c$ rejects more EMG burst noise but lengthens the delay between intent and grip. $f_c = 0$ is the limit of pure averaging.