{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Numerical signal processing\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/numerical-signal-processing). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "Numerical signal processing turns sampled measurements into new arrays: smoothed signals,\n", "estimated derivatives, spectra, detected events, or lower-rate records. Every operation\n", "changes what information is emphasized or discarded. A smooth curve is not automatically\n", "more accurate than the raw data; it may be delayed, attenuated, or distorted near an edge.\n", "\n", "## Begin with the sampling contract\n", "\n", "For uniformly spaced samples $x[n]$\n", "with time interval $\\Delta t$, the\n", "sample rate is\n", "\n", "\n", "$$\n", "f_s=\\frac{1}{\\Delta t}\n", "$$\n", "\n", "\n", "The sample at index $n$ represents time\n", "$t_n=t_0+n\\Delta t$. Without\n", "$t_0$ and\n", "$\\Delta t$, an array has no physical\n", "frequency scale. Uneven timestamps cannot be silently treated as uniform; they require\n", "resampling with an error model or a method designed for irregular observations.\n", "\n", "Before filtering, identify missing values, saturation, clipped intervals, unit changes,\n", "and sensor calibration. A filter cannot recover amplitudes lost when a detector clipped.\n", "Replacing missing values by zero adds an artificial impulse unless zero genuinely means\n", "the measured physical quantity was zero.\n", "\n", "## Finite impulse response convolution\n", "\n", "A finite impulse response filter with coefficients\n", "$h[m]$ produces\n", "\n", "\n", "$$\n", "y[n]=\\sum_{m=0}^{M-1}h[m]x[n-m]\n", "$$\n", "\n", "\n", "where $M$ is the number of coefficients.\n", "This causal indexing uses the current and previous samples. A centered smoother uses\n", "future and past indices instead and is noncausal in live processing, although it is\n", "possible for an already recorded signal." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "samples = np.array([0.0, 0.0, 1.0, 0.0, 0.0, 0.0])\n", "kernel = np.ones(3) / 3\n", "filtered = np.convolve(samples, kernel, mode=\"full\")[: len(samples)]\n", "print(\"Causal moving average with zero padding:\", filtered)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "The filter must define what happens when\n", "$n-m$ lies outside the record.\n", "Zero padding, constant extension, reflection, periodic wrapping, and shortened kernels\n", "produce different edge values. None is universally correct.\n", "\n", "## Moving averages\n", "\n", "The length-three causal moving average has\n", "\n", "\n", "$$\n", "h[0]=h[1]=h[2]=\\frac13\n", "$$\n", "\n", "\n", "Its coefficients sum to one, so after start-up it preserves a constant input under\n", "compatible edge handling. An impulse reveals the coefficients directly. A step response\n", "shows the transition spread over several samples.\n", "\n", "Its frequency response is found by applying the filter to a complex sinusoid:\n", "\n", "\n", "$$\n", "H(e^{i\\Omega})=\\sum_{m=0}^{M-1}h[m]e^{-i\\Omega m}\n", "$$\n", "\n", "\n", "Here $\\Omega=2\\pi f/f_s$\n", "is radians per sample, not radians per second. The magnitude\n", "$|H|$ tells how amplitude\n", "changes, while phase tells how timing changes.\n", "\n", "A symmetric odd-length FIR filter used with centered indexing has linear phase and a\n", "constant delay of $(M-1)/2$\n", "when implemented causally. Subtracting this delay in a plot does not make a live filter\n", "predict the future; it only realigns an offline result.\n", "\n", "## Smoothing and differentiation pull in opposite directions\n", "\n", "A centered finite-difference derivative is\n", "\n", "\n", "$$\n", "\\left.\\frac{dx}{dt}\\right|_{t_n}\\approx\\frac{x[n+1]-x[n-1]}{2\\Delta t}\n", "$$\n", "\n", "\n", "Differentiation amplifies high-frequency noise because rapid sample-to-sample changes are\n", "divided by a small time interval. Smoothing first can reduce noise but also attenuates\n", "real fast motion and shifts or broadens events. Report the derivative stencil and any\n", "prior filter as one processing chain.\n", "\n", "Do not repeatedly filter until a desired-looking curve appears. Select parameters using a\n", "physical bandwidth, calibration data, cross-validation, or an explicit noise model, then\n", "test sensitivity to reasonable alternatives.\n", "\n", "## Infinite impulse response filters\n", "\n", "A recursive filter uses previous outputs as well as inputs. A simple first-order\n", "low-pass update is\n", "\n", "\n", "$$\n", "y[n]=(1-\\alpha)y[n-1]+\\alpha x[n],\\qquad 0<\\alpha\\le1\n", "$$\n", "\n", "\n", "Small $\\alpha$ gives stronger\n", "smoothing and a slower response. The initial value\n", "$y[-1]$ matters. This\n", "filter's impulse response continues indefinitely, and changing the recursion coefficients\n", "can make an IIR filter unstable even when every input sample is bounded." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "alpha, previous = 0.2, 0.0\n", "output = []\n", "for sample in np.ones(10):\n", " current = (1 - alpha) * previous + alpha * sample\n", " output.append(current)\n", " previous = current\n", "print(\"Smoothed step:\", output)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "Forward filtering is causal and introduces phase delay. Applying a filter forward and\n", "backward can cancel phase delay for offline data, but it squares the magnitude response\n", "and uses future samples. Label it as noncausal.\n", "\n", "## Filtering in the frequency domain\n", "\n", "The convolution theorem permits filtering by transforming the input and kernel,\n", "multiplying corresponding complex bins, and applying the inverse transform:\n", "\n", "\n", "$$\n", "Y_k=X_kH_k\n", "$$\n", "\n", "\n", "For finite DFT arrays, this computes circular convolution. To obtain ordinary linear\n", "convolution of lengths $N$ and\n", "$M$, zero-pad to at least\n", "$N+M-1$ samples before\n", "transforming. Otherwise the end wraps into the beginning.\n", "\n", "An ideal brick-wall frequency mask has an abrupt cutoff and an infinitely long oscillatory\n", "impulse response. On a finite record it can create ringing. A gradual, physically justified\n", "transition band often behaves better, but its attenuation and retained bandwidth must be\n", "measured rather than inferred from appearance.\n", "\n", "## Resampling and alias protection\n", "\n", "Reducing sample rate discards frequency range. Before keeping every\n", "$D$th sample, low-pass filter\n", "the signal so retained content lies below the new Nyquist frequency:\n", "\n", "\n", "$$\n", "f_{\\mathrm{new}}=\\frac{f_s}{D},\\qquad f_{\\mathrm{Nyquist,new}}=\\frac{f_s}{2D}\n", "$$" ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "# Offline ideal low-pass filtering of a periodic signal before decimation.\n", "rate, factor = 1000.0, 4\n", "time = np.arange(1000) / rate\n", "signal = np.sin(2 * np.pi * 20 * time) + 0.5 * np.sin(2 * np.pi * 200 * time)\n", "frequencies = np.fft.rfftfreq(len(signal), d=1 / rate)\n", "spectrum = np.fft.rfft(signal)\n", "spectrum[frequencies >= 0.45 * rate / factor] = 0\n", "anti_aliased = np.fft.irfft(spectrum, n=len(signal))\n", "decimated = anti_aliased[::factor]\n", "print(\"New sample rate (Hz):\", rate / factor, \"samples:\", len(decimated))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "Filtering after decimation is too late: aliased high-frequency content has already become\n", "indistinguishable from lower frequencies.\n", "\n", "Upsampling by inserting zeros creates spectral images that an interpolation filter must\n", "remove. It does not invent new measured bandwidth. More output points can make a graph\n", "look smoother without adding experimental information.\n", "\n", "## Normalization, gain, and units\n", "\n", "For a smoothing filter intended to preserve DC, verify\n", "\n", "\n", "$$\n", "\\sum_{m=0}^{M-1}h[m]=1\n", "$$\n", "\n", "\n", "Other filters need not sum to one. A discrete derivative kernel, for example, should sum\n", "to zero so a constant input produces zero. Its coefficients include inverse-time units\n", "through division by $\\Delta t$.\n", "\n", "If input samples have volts, metres, or pascals, state the output units after every stage.\n", "Spectral amplitudes also depend on DFT and window normalization. Unit and gain checks catch\n", "many seemingly plausible scaling errors.\n", "\n", "## Verification fixtures\n", "\n", "Use a constant to test DC gain, an impulse to expose the complete impulse response, a step\n", "to reveal delay and ringing, and bin-aligned sinusoids to measure gain and phase at known\n", "frequencies. A sum of two sinusoids can confirm that a linear filter processes each\n", "component independently." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "constant = np.full(20, 5.0)\n", "filtered_constant = np.convolve(constant, kernel, mode=\"valid\")\n", "assert np.allclose(filtered_constant, 5.0)\n", "impulse = np.zeros(10)\n", "impulse[0] = 1\n", "assert np.allclose(np.convolve(impulse, kernel)[:3], kernel)\n", "print(\"Constant and impulse checks passed.\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "Compare time-domain convolution with a properly padded frequency-domain calculation.\n", "Ensure every output is finite. Record raw and processed arrays, coefficients, edge policy,\n", "sample rate, software revision, and error metrics. If the filter supports a physical\n", "claim, repeat the analysis over a defensible parameter range and report whether the claim\n", "survives.\n", "\n", "Linear filters cannot undo clipping, distinguish aliased frequencies, or separate two\n", "sources without additional assumptions. Adaptive and nonlinear filters require their own\n", "validation. Signal processing can improve an estimate when its assumptions match the\n", "measurement, but it cannot replace a measurement model.\n", "\n", "## A worked Python notebook\n", "\n", "Study a first-order smoother that starts at zero and receives a constant input of one. The question for this calculation is precise: how does **response after ten unit-step samples**\n", "change when we change **smoothing factor**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "The factor controls both noise smoothing and delay; it is not a physical damping coefficient without a model linking them. Keep that sentence beside the results. A formula can return a number outside\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "0.01 to 1 dimensionless. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for smoothing factor, expressed in dimensionless. The output is\n", "response after ten unit-step samples, expressed in dimensionless. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = 1 - \\left(1 - q\\right)^{10}\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 0.2 dimensionless, the expression gives 0.892626 dimensionless. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"1-(1-q)**10\", locals={\"q\": q})\n", "reference_input = 0.2\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"dimensionless\")\n", "print(\n", " \"Predicted response after ten unit-step samples:\", reference_output, \"dimensionless\"\n", ")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "0.0956179 dimensionless. The midpoint, at 0.505 dimensionless, gives\n", "0.999117 dimensionless. The upper endpoint gives 1 dimensionless.\n", "Read these together with the assumptions: they describe different choices of smoothing factor\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.01, 0.505, 1.0])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} dimensionless -> {output_value:.6g} dimensionless\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows smoothing factor; the vertical axis shows response after ten unit-step samples. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "input_grid = np.linspace(0.01, 1.0, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Smoothing factor (dimensionless)\")\n", "ax.set_ylabel(\"Response after ten unit-step samples (dimensionless)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "response after ten unit-step samples change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 0.2 dimensionless is\n", "1.34218 in those ratio units. Locally, the output increases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = 10 \\left(1 - q\\right)^{9}\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 0.20495000000000002\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"dimensionless\")\n", "print(\"Full expression:\", full_prediction, \"dimensionless\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"dimensionless\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "Here the changed input is 0.20495 dimensionless. The tangent estimate is\n", "0.89927 dimensionless, while the full expression gives\n", "0.899088 dimensionless. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-019", "metadata": {}, "source": [ "input_span = 0.99\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-020", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "smoothing factor inside the stated interval, predict whether response after ten unit-step samples will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "Finally, return to the physical limit stated at the start: The factor controls both noise smoothing and delay; it is not a physical damping coefficient without a model linking them. Describe one change\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of dimensionless:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-021", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(1.0) - evaluate(0.01))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.01, 1.0, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-022", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "\n", "- [Benjamin Crowell, *Light and Matter*, sampled data, waves, and Fourier analysis](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, “Music and Noise”](https://physics.info/music/).\n", "- [Physics LibreTexts, *The Physics of Waves*, Chapter 10, “Signals and Fourier Analysis”](https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/The_Physics_of_Waves_%28Goergi%29/10%3A_Signals_and_Fourier_Analysis).\n", "- [Physics LibreTexts, “Fourier Analysis and Synthesis”](https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Waves%3A_An_Interactive_Tutorial_%28Forinash_and_Christian%29/2%3A_Combining_Waves/2.5%3A_Fourier_Analysis_and_Synthesis)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "numerical-signal-processing.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/numerical-signal-processing", "source_sha256": "373c1825c5cd26f8acb516bc433f2b7ba2041e41b44b2fc311024f6c19d43e00" } }, "nbformat": 4, "nbformat_minor": 5 }