Update Content - 2022-03-01
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title = "Actuators"
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author = ["Thomas Dehaeze"]
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author = ["Dehaeze Thomas"]
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draft = false
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@@ -17,19 +17,14 @@ Links to specific actuators:
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For vibration isolation:
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- In ([Ito and Schitter 2016](#orge96c061)), the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{<relref "ito16_compar_class_high_precis_actuat.md#" >}}))
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- In <ito16_compar_class_high_precis_actuat>, the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{<relref "ito16_compar_class_high_precis_actuat.md#" >}}))
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## Brush-less DC Motor {#brush-less-dc-motor}
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- ([Yedamale 2003](#org9fa946a))
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- <yedamale03_brush_dc_bldc_motor_fundam>
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<https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html>
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## Bibliography {#bibliography}
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<a id="orge96c061"></a>Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” _IEEE/ASME Transactions on Mechatronics_ 21 (2):1169–78. <https://doi.org/10.1109/tmech.2015.2478658>.
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<a id="org9fa946a"></a>Yedamale, Padmaraja. 2003. “Brushless Dc (BLDC) Motor Fundamentals.” _Microchip Technology Inc_ 20:3–15.
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## [Stepper Motor]({{<relref "stepper_motor.md#" >}}) {#stepper-motor--stepper-motor-dot-md}
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content/zettels/decimation.md
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content/zettels/decimation.md
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title = "Decimation"
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author = ["Dehaeze Thomas"]
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draft = false
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Tags
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: [Digital Signal Processing]({{< relref "digital_signal_processing.md" >}})
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<div class="definition">
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Decimation is the two-step process of low pass filtering followed by and operation known as downsampling.
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</div>
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We can downsample a sequence of sampled signal values by a factor of \\(M\\) by retaining every Mth sample and discarding all the remaining samples.
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Relative to the original sample rate \\(f\_{s,\text{old}}\\), the sample rate of the downsampled sequence is:
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\begin{equation}
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f\_{s,\text{new}} = \frac{f\_{s,\text{old}}}{M}
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\end{equation}
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<div class="exampl">
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For example, assume that an analog sinewave has been sampled to produce \\(x\_{\text{old}}(n)\\).
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The downsampled sequence is:
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\\[ x\_{\text{new}}(m) = x\_{\text{old}}(Nm) \\]
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where \\(M=3\\), the result is shown in Figure [1](#figure--fig:decimation-example).
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<a id="figure--fig:decimation-example"></a>
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{{< figure src="/ox-hugo/decimation_example.png" caption="<span class=\"figure-number\">Figure 1: </span>Sample rate conversion: (a) original sequence; (b) downsampled by \\(M=3\\) sequence" >}}
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</div>
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The spectral implications of downsampling are what we should expect as shown in Figure
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<a id="figure--fig:decimation-spectral-aliasing"></a>
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{{< figure src="/ox-hugo/decimation_spectral_aliasing.png" caption="<span class=\"figure-number\">Figure 2: </span>Decimation by a factor of three: (a) spectrum of original \\(x\_{\text{old}}(n)\\) signal; (b) spectrum after downsampling by three." >}}
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There is a limit to the amount of downsampling that can be performed relative to the bandwidth \\(B\\) of the original signal.
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We must ensure that \\(f\_{s,\text{new}} > 2B\\) to present overlapped spectral replications (aliasing errors) after downsampling.
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If a decimation application requires \\(f\_{s,\text{new}}\\) to be less than \\(2B\\), then \\(x\_{\text{old}}(n)\\) must be low pass filtered before the downsampling process if performed.
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## Two Stage Decimation {#two-stage-decimation}
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When the desired decimation factor \\(M\\) is larger, say \\(M > 20\\), there is an important feature of the filter / decimation process to keep in mind.
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Significant low pass filter computational savings may be obtained by implementing the two-stage decimation, shown in Figure [3](#figure--fig:decimation-two-stages) (b).
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<a id="figure--fig:decimation-two-stages"></a>
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{{< figure src="/ox-hugo/decimation_two_stages.png" caption="<span class=\"figure-number\">Figure 3: </span>Decimation: (a) single-stage; (b) two-stage" >}}
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The question is: "Given a desired total downsampling factor \\(M\\), what should be the values of \\(M\_1\\) and \\(M\_2\\) to minimize the number of taps in low-pass filters \\(\text{LPF}\_1\\) and \\(\text{LPF}\_2\\)"?
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For two stage decimation, the optimum value for \\(M\_1\\) is:
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\begin{equation} \label{eq:M1opt}
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M\_{1,\text{opt}} \approx 2 M \cdot \frac{1 - \sqrt{MF/(2-F)}}{2 - F(M+1)}
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\end{equation}
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where \\(F\\) is the ratio of single-stage low pass filter's transition region width to that filter's stop-band frequency:
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\begin{equation}
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F = \frac{f\_{\text{stop}} - B^\prime}{f\_{\text{stop}}}
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\end{equation}
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After using Eq. <eq:M1opt> to determine the optimum first downsampling factor, and setting \\(M\_1\\) equal to the integer sub-multiple of \\(M\\) that is closest to \\(M\_{1,\text{opt}}\\), the second downsampling factor is:
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\begin{equation} \label{eq:M2\_from\_M1}
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M\_2 = \frac{M}{M\_1}
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\end{equation}
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<div class="exampl">
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Let's assume we have an \\(x\_{\text{old}}(n)\\) input signal arriving at a sample rate of \\(400\\,kHz\\), and we must decimate that signal by a factor of \\(M=100\\) to obtain a final sample rate of \\(4\\,kHz\\).
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Also, let's assume the base-band frequency range of interest is from \\(0\\) to \\(B^\prime = 1.8\\,kHz\\), and we want \\(60\\,dB\\) of filter stop-band attenuation.
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A single stage decimation low-pass filter's frequency response is shown in Figure [4](#figure--fig:decimation-two-stage-example) (a).
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The number of taps \\(N\\) required for a single-stage decimation would be:
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\begin{equation}
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N = \frac{\text{Atten}}{22 (f\_{\text{stop}} - f\_{\text{pass}})} = \frac{60}{22(2.2/400 - 1.8/400)} = 2727
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\end{equation}
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which is way too large for practical implementation.
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To reduce the number of necessary filter taps, we can partition the decimation problem into two stages.
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With \\(M = 100\\), \\(F = (2200-1800)/2200\\), Eq. <eq:M1opt> yields \\(M\_{1,\text{opt}} = 26.4\\).
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The integer sub-multiple of 100 closest to \\(26.4\\) is \\(25\\), so we set \\(M\_1 = 25\\).
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Next, from Eq. <eq:M2_from_M1>, \\(M\_2 = 4\\) is found.
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The first low pass filter has a pass-band cutoff frequency of \\(1.8\\,kHz\\) and its stop-band is \\(400/25 - 1.8 = 14.2\\,kHz\\) (Figure [4](#figure--fig:decimation-two-stage-example) (d)).
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The second low pass filter has a pass-band cutoff frequency of \\(1.8\\,kHz\\) and its stop-band is \\(4-1.8 = 2.2\\,kHz\\).
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The total number of required taps is:
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\begin{equation}
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N\_{\text{total}} = N\_{\text{LPF}\_1} + N\_{\text{LPF}\_2} = \frac{60}{22(14.2/400-1.8/400)} + \frac{60}{22(2.2/16 - 1.8/16)} \approx 197
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\end{equation}
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Which is much more efficient that the single stage decimation.
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<a id="figure--fig:decimation-two-stage-example"></a>
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{{< figure src="/ox-hugo/decimation_two_stage_example.png" caption="<span class=\"figure-number\">Figure 4: </span>Two stage decimation: (a) single-stage filter response; (b): decimation by 100; (c) spectrum of original signal; (d) output spectrum of the \\(M=25\\) down-sampler; (e) output spectrum of the \\(M=4\\) down-sampler." >}}
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</div>
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## References: {#references}
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<lyons11_under_digit_signal_proces>
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title = "Power Spectral Density"
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author = ["Thomas Dehaeze"]
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author = ["Dehaeze Thomas"]
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draft = false
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+++
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Tags
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: [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}})
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: [Signal to Noise Ratio]({{<relref "signal_to_noise_ratio.md#" >}})
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Tutorial about Power Spectral Density is accessible [here](https://research.tdehaeze.xyz/spectral-analysis/).
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A good article about how to use the `pwelch` function with Matlab ([Schmid 2012](#org7c6692c)).
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A good article about how to use the `pwelch` function with Matlab <schmid12_how_to_use_fft_matlab>.
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## Parseval's Theorem - Linking the Frequency and Time domain {#parseval-s-theorem-linking-the-frequency-and-time-domain}
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## Bibliography {#bibliography}
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For non-periodic finite duration signals, the energy in the time domain is described by:
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<a id="org7c6692c"></a>Schmid, Hanspeter. 2012. “How to Use the FFT and Matlab’s Pwelch Function for Signal and Noise Simulations and Measurements.” _Institute of Microelectronics_.
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\begin{equation}
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\text{Energy} = \int\_{-\infty}^\infty x(t)^2 dt
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\end{equation}
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Parseval's Theorem states that energy in the time domain equals energy in the frequency domain:
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\begin{equation}
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\text{Energy} = \int\_{-\infty}^{\infty} x(t)^2 dt = \int\_{-\infty}^{\infty} |X(f)|^2 df
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\end{equation}
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where \\(X(f)\\) is the Fourier transform of the time signal \\(x(t)\\):
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\begin{equation}
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X(f) = \int\_{-\infty}^{\infty} x(t) e^{-2\pi j f t} dt
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\end{equation}
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## Power Spectral Density function (PSD) {#power-spectral-density-function--psd}
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The power distribution over frequency of a time signal \\(x(t)\\) is described by the PSD denoted the \\(S\_x(f)\\).
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A PSD is a power density function with units \\([\text{SI}^2/Hz]\\), meaning that the area underneath the PSD curve equals the power (units \\([\text{SI}^2]\\)) of the signal (SI is the unit of the signal, e.g. \\(m/s\\)).
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Using the definition of signal power \\(\bar{x^2}\\) and Parseval's theorem, we can link power in the time domain with power in the frequency domain:
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\begin{equation}
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\text{power} = \lim\_{T \to \infty} \frac{1}{2T} \int\_{-T}^{T} x\_T(t)^2 dt = \lim\_{T \to \infty} \frac{1}{2T} \int\_{-\infty}^{\infty} |X\_T(f)|^2 df = \int\_{-\infty}^{\infty} \left( \lim\_{T \to \infty} \frac{|X\_T(f)|^2}{2T} \right) df
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\end{equation}
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where \\(X\_T(f)\\) denotes the Fourier transform of \\(x\_T(t)\\), which equals \\(x(t)\\) on the interval \\(-T \le t \le T\\) and is zero outside this interval.
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This term is referred to as the two-sided spectral density:
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\begin{equation}
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S\_{x,two} (f) = \lim\_{T \to \infty} \frac{|X\_T(f)|^2}{2T}, \quad -\infty \le f \le \infty
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\end{equation}
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In practice, the **one sided PSD** is used, which is only defined on the positive frequency axis but also contains all the power.
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It is defined as:
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\begin{equation}
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S\_{x}(f) = \lim\_{T \to \infty} \frac{|X\_T(f)|^2}{T}, \quad 0 \le f \le \infty
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\end{equation}
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For discrete time signals, the one-sided PSD estimate is defined as:
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\begin{equation}
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\hat{S}(f\_k) = \frac{|X\_L(f\_k)|^2}{L T\_s}
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\end{equation}
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where \\(L\\) equals the number of time samples and \\(T\_s\\) the sample time, \\(X\_L(f\_k)\\) is the N-point discrete Fourier Transform of the discrete time signal \\(x\_L[n]\\) containing \\(L\\) samples:
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\begin{equation}
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X\_L(f\_k) = \sum\_{n = 0}^{N-1} x\_L[n] e^{-j 2 \pi k n/N}
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\end{equation}
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## Matlab Code for computing the PSD and CPS {#matlab-code-for-computing-the-psd-and-cps}
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Let's compute the PSD of a signal by "hand".
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The signal is defined below.
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```matlab
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%% Signal generation
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T_s = 1e-3; % Sampling Time [s]
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t = T_s:T_s:100; % Time vector [s]
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L = length(t);
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x = lsim(1/(1 + s/2/pi/5), randn(1, L), t);
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```
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The computation is performed using the `fft` function.
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```matlab
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%% Parameters
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T_r = L*T_s; % signal time range
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d_f = 1/T_r; % width of frequency grid
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F_s = 1/T_s; % sample frequency
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F_n = F_s/2; % Nyquist frequency
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F = [0:d_f:F_n]; % one sided frequency grid
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% Discrete Time Fourier Transform Wxx
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Wxx = fft(x - mean(x))/L;
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% Two-sided Power Spectrum Pxx [SI^2]
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Pxx = Wxx.*conj(Wxx);
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% Two-sided Power Spectral Density Sxx_t [SI^2/Hz]
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Sxx_t = Pxx/d_f;
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% One-sided Power Spectral Density Sxx_o [SI^2/Hz] defined on F
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Sxx_o = 2*Sxx_t(1:L/2+1);
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```
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The result is shown in Figure [1](#org41c99c6).
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<a id="org41c99c6"></a>
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{{< figure src="/ox-hugo/psd_manual_example.png" caption="Figure 1: Amplitude Spectral Density with manual computation" >}}
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This can also be done using the `pwelch` function which integrated a "window" that permits to do some averaging.
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```matlab
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%% Computation using pwelch function
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[pxx, f] = pwelch(x, hanning(ceil(5/T_s)), [], [], 1/T_s);
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```
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The comparison of the two method is shown in Figure [2](#orge7a31a8).
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<a id="orge7a31a8"></a>
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{{< figure src="/ox-hugo/psd_comp_pwelch_manual_example.png" caption="Figure 2: Amplitude Spectral Density with manual computation" >}}
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19
content/zettels/stepper_motor.md
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19
content/zettels/stepper_motor.md
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@@ -0,0 +1,19 @@
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+++
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title = "Stepper Motor"
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author = ["Dehaeze Thomas"]
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draft = false
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+++
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Tags
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:
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## 2 phase VS 5 phase stepper motor {#2-phase-vs-5-phase-stepper-motor}
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<https://www.orientalmotor.com/stepper-motors/technology/2-phase-vs-5-phase-stepper-motors.html>
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## Errors {#errors}
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For a two phase stepper motor, there are (typically) 200 steps per revolution.
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Errors with a period of 200 period/revolution can be expected.
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