**This page shows how to use the LM741 op-amp model file in LTSpice.** This is surprisingly un-intuitive, but is a good thing to know how to do. Model files can often be downloaded by vendor sites, but LTSpice only comes pre-loaded with models of common LT components.

I found `LM741.MOD`

available on the TI’s LM741 product page.

Save it wherever you want, but you will need to know the full path to this file later.

Open the model file in a text editor and look for the line starting with `.SUBCKT`

. The top of LM741.MOD looks like this:

```
* connections: non-inverting input
* | inverting input
* | | positive power supply
* | | | negative power supply
* | | | | output
* | | | | |
* | | | | |
.SUBCKT LM741/NS 1 2 99 50 28
```

The last line tells us the name of this model’s sub-circuit is `LM741/NS`

Click the “.op” button on the toolbar, then add `.include`

followed by the full path to the model file. After clicking OK place the text somewhere on your LTSpice circuit diagram.

We know the part we are including is a 5-pin op-amp, so we can start by placing a generic component. Notice the description says *you must give the value a name and include this file*. We will do this in the next step.

Right-click the op-amp and update its `Value`

to match the name of the subcircuit we read from the model file earlier.

Your new component will run using the properties of the model you downloaded.

**Fitting an exponential curve to data is a common task** and in this example we’ll use Python and SciPy to determine parameters for a curve fitted to arbitrary X/Y points. You can follow along using the fit.ipynb Jupyter notebook.

```
import numpy as np
import scipy.optimize
import matplotlib.pyplot as plt
xs = np.arange(12) + 7
ys = np.array([304.08994, 229.13878, 173.71886, 135.75499,
111.096794, 94.25109, 81.55578, 71.30187,
62.146603, 54.212032, 49.20715, 46.765743])
plt.plot(xs, ys, '.')
plt.title("Original Data")
```

**To fit an arbitrary curve** we must first define it as a function. We can then call `scipy.optimize.curve_fit`

which will tweak the arguments (using arguments we provide as the starting parameters) to best fit the data. In this example we will use a single exponential decay function.

```
def monoExp(x, m, t, b):
return m * np.exp(-t * x) + b
```

**In biology / electrophysiology ***biexponential* functions are often used to separate fast and slow components of exponential decay which may be caused by different mechanisms and occur at different rates. In this example we will only fit the data to a method with a exponential component (a *monoexponential* function), but the idea is the same.

```
# perform the fit
p0 = (2000, .1, 50) # start with values near those we expect
params, cv = scipy.optimize.curve_fit(monoExp, xs, ys, p0)
m, t, b = params
sampleRate = 20_000 # Hz
tauSec = (1 / t) / sampleRate
# determine quality of the fit
squaredDiffs = np.square(ys - monoExp(xs, m, t, b))
squaredDiffsFromMean = np.square(ys - np.mean(ys))
rSquared = 1 - np.sum(squaredDiffs) / np.sum(squaredDiffsFromMean)
print(f"R² = {rSquared}")
# plot the results
plt.plot(xs, ys, '.', label="data")
plt.plot(xs, monoExp(xs, m, t, b), '--', label="fitted")
plt.title("Fitted Exponential Curve")
# inspect the parameters
print(f"Y = {m} * e^(-{t} * x) + {b}")
print(f"Tau = {tauSec * 1e6} µs")
```

```
Y = 2666.499 * e^(-0.332 * x) + 42.494
Tau = 150.422 µs
R² = 0.999107330342064
```

**We can use the calculated parameters to extend this curve** to any position by passing X values of interest into the function we used during the fit.

**The value at time 0** is simply `m + b`

because the exponential component becomes e^(0) which is 1.

```
xs2 = np.arange(25)
ys2 = monoExp(xs2, m, t, b)
plt.plot(xs, ys, '.', label="data")
plt.plot(xs2, ys2, '--', label="fitted")
plt.title("Extrapolated Exponential Curve")
```

**What if we know our data decays to 0?** It’s not best to fit to an exponential decay function that lets the `b`

component be whatever it wants. Indeed, our fit from earlier calculated the ideal `b`

to be `42.494`

but what if we know it should be `0`

? The solution is to fit using an exponential function where `b`

is constrained to 0 (or whatever value you know it to be).

```
def monoExpZeroB(x, m, t):
return m * np.exp(-t * x)
# perform the fit using the function where B is 0
p0 = (2000, .1) # start with values near those we expect
paramsB, cv = scipy.optimize.curve_fit(monoExpZeroB, xs, ys, p0)
mB, tB = paramsB
sampleRate = 20_000 # Hz
tauSec = (1 / tB) / sampleRate
# inspect the results
print(f"Y = {mB} * e^(-{tB} * x)")
print(f"Tau = {tauSec * 1e6} µs")
# compare this curve to the original
ys2B = monoExpZeroB(xs2, mB, tB)
plt.plot(xs, ys, '.', label="data")
plt.plot(xs2, ys2, '--', label="fitted")
plt.plot(xs2, ys2B, '--', label="zero B")
```

```
Y = 1245.580 * e^(-0.210 * x)
Tau = 237.711 µs
```

**The curves produced are very different** at the extremes (especially when time is 0), even though they appear to both fit the data points nicely. Which curve is more accurate? That depends on your application. A hint can be gained by inspecting the time constants of these two curves.

Parameter |
Fitted B |
Fixed B |

m |
2666.499 |
1245.580 |

t |
0.332 |
0.210 |

Tau |
150.422 µs |
237.711 µs |

b |
42.494 |
0 |

**By inspecting Tau** I can gain insight into which method may be better for me to use in my application. I expect Tau to be near 250 µs, leading me to trust the fixed-B method over the fitted B method. Choosing the correct method has great implications on the value of `m`

(which is also the value of the curve when time is 0).

How to apply low-pass, high-pass, and band-pass filters with Python

**This page describes how to perform low-pass, high-pass, and band-pass filtering in Python.** I favor SciPy’s `filtfilt`

function because the filtered data it produces is the same length as the source data and it has no phase offset, so the output always aligns nicely with the input. The `sosfiltfilt`

function is even more convenient because it consumes filter parameters as a single object which makes them easier work with.

```
import numpy as np
import scipy.signal
import scipy.io.wavfile
import matplotlib.pyplot as plt
def lowpass(data: np.ndarray, cutoff: float, sample_rate: float, poles: int = 5):
sos = scipy.signal.butter(poles, cutoff, 'lowpass', fs=sample_rate, output='sos')
filtered_data = scipy.signal.sosfiltfilt(sos, data)
return filtered_data
# Load sample data from a WAV file
sample_rate, data = scipy.io.wavfile.read('ecg.wav')
times = np.arange(len(data))/sample_rate
# Apply a 50 Hz low-pass filter to the original data
filtered = lowpass(data, 50, sample_rate)
```

```
# Code used to display the result
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 3), sharex=True, sharey=True)
ax1.plot(times, data)
ax1.set_title("Original Signal")
ax1.margins(0, .1)
ax1.grid(alpha=.5, ls='--')
ax2.plot(times, filtered)
ax2.set_title("Low-Pass Filter (50 Hz)")
ax2.grid(alpha=.5, ls='--')
plt.tight_layout()
plt.show()
```

```
import numpy as np
import scipy.signal
import scipy.io.wavfile
import matplotlib.pyplot as plt
def highpass(data: np.ndarray, cutoff: float, sample_rate: float, poles: int = 5):
sos = scipy.signal.butter(poles, cutoff, 'highpass', fs=sample_rate, output='sos')
filtered_data = scipy.signal.sosfiltfilt(sos, data)
return filtered_data
# Load sample data from a WAV file
sample_rate, data = scipy.io.wavfile.read('ecg.wav')
times = np.arange(len(data))/sample_rate
# Apply a 20 Hz high-pass filter to the original data
filtered = highpass(data, 20, sample_rate)
```

```
# Code used to display the result
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 3), sharex=True, sharey=True)
ax1.plot(times, data)
ax1.set_title("Original Signal")
ax1.margins(0, .1)
ax1.grid(alpha=.5, ls='--')
ax2.plot(times, filtered)
ax2.set_title("High-Pass Filter (20 Hz)")
ax2.grid(alpha=.5, ls='--')
plt.tight_layout()
plt.show()
```

```
import numpy as np
import scipy.signal
import scipy.io.wavfile
import matplotlib.pyplot as plt
def bandpass(data: np.ndarray, edges: list[float], sample_rate: float, poles: int = 5):
sos = scipy.signal.butter(poles, edges, 'bandpass', fs=sample_rate, output='sos')
filtered_data = scipy.signal.sosfiltfilt(sos, data)
return filtered_data
# Load sample data from a WAV file
sample_rate, data = scipy.io.wavfile.read('ecg.wav')
times = np.arange(len(data))/sample_rate
# Apply a 10-50 Hz high-pass filter to the original data
filtered = bandpass(data, [10, 50], sample_rate)
```

```
# Code used to display the result
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 3), sharex=True, sharey=True)
ax1.plot(times, data)
ax1.set_title("Original Signal")
ax1.margins(0, .1)
ax1.grid(alpha=.5, ls='--')
ax2.plot(times, filtered)
ax2.set_title("Band-Pass Filter (10-50 Hz)")
ax2.grid(alpha=.5, ls='--')
plt.tight_layout()
plt.show()
```

This code evaluates the same signal low-pass filtered using different cutoff frequencies:

```
import numpy as np
import scipy.signal
import scipy.io.wavfile
import matplotlib.pyplot as plt
# Load sample data from a WAV file
sample_rate, data = scipy.io.wavfile.read('ecg.wav')
times = np.arange(len(data))/sample_rate
# Plot the original signal
plt.plot(times, data, '.-', alpha=.5, label="original signal")
# Plot the signal low-pass filtered using different cutoffs
for cutoff in [10, 20, 30, 50]:
sos = scipy.signal.butter(5, cutoff, 'lowpass', fs=sample_rate, output='sos')
filtered = scipy.signal.sosfiltfilt(sos, data)
plt.plot(times, filtered, label=f"low-pass {cutoff} Hz")
plt.legend()
plt.grid(alpha=.5, ls='--')
plt.axis([0.35, 0.5, None, None])
plt.show()
```

**Artifacts may appear in the smooth signal if the first or last data point differs greatly from their adjacent points.** This is because, in an effort to ensure the filtered signal length is the same as the input signal, the input signal is “padded” with data on each side prior to filtering. The default behavior is to pad the data by duplicating the first and last data points, but this causes artifacts in the smoothed signal if the first or last points contain an extreme value. An alternative strategy is *Gustafsson’s Method*, described in a 1996 paper by Fredrik Gustafsson in which “initial conditions are chosen for the forward and backward passes so that the forward-backward filter gives the same result as the backward-forward filter.” Interestingly, the original publication demonstrates the method by filtering noise out of an ECG recording.

```
import numpy as np
import scipy.signal
import scipy.io.wavfile
import matplotlib.pyplot as plt
# Load sample data from a WAV file
sample_rate, data = scipy.io.wavfile.read('ecg.wav')
times = np.arange(len(data))/sample_rate
# Isolate a small portion of data to inspect
segment = data[350:400]
# Create a 5-pole low-pass filter with an 80 Hz cutoff
b, a = scipy.signal.butter(5, 80, fs=sample_rate)
# Apply the filter using the default edge method (padding)
filtered_pad = scipy.signal.filtfilt(b, a, segment)
# Apply the filter using Gustafsson's method
filtered_gust = scipy.signal.filtfilt(b, a, segment, method="gust")
```

```
# Display the Results
plt.plot(segment, '.-', alpha=.5, label="data")
plt.plot(filtered_pad, 'k--', label="Default (Padding)")
plt.plot(filtered_gust, 'k', label="Gustafsson's Method")
plt.legend()
plt.grid(alpha=.5, ls='--')
plt.title("Padded Data vs. Gustafsson’s Method")
plt.show()
```

**An alternative strategy to low-pass a signal is to use convolution.** In this method you create a kernel (typically a bell-shaped curve) and *convolve* the kernel with the signal. The wider the window is the smoother the output signal will be. Also, the window must be normalized so its sum is 1 to preserve the amplitude of the input signal. Note that this method exclusively uses NumPy and does not require SciPy.

**There are different for handling data at the edges of the signal,** but setting `mode`

to `valid`

deletes insufficiently filtered points at the edges to produce an output signal that is fully filtered but slightly shorter than the input signal. See `numpy.convolve`

documentation for additional information.

**The kernel shape affects the spectral properties of the filter.** Commonly called *window functions*, these different shapes produce filtered signals with different frequency response characteristics. The Hanning window is preferred for most general purpose signal processing applications. See FftSharp for additional information about the pros and cons of common window functions.

```
import numpy as np
import scipy.io.wavfile
import matplotlib.pyplot as plt
# Load sample data from a WAV file
sample_rate, data = scipy.io.wavfile.read('ecg.wav')
times = np.arange(len(data))/sample_rate
# create a Hanning kernel 1/50th of a second wide
kernel_width_seconds = 1.0/50
kernel_size_points = int(kernel_width_seconds * sample_rate)
kernel = np.hanning(kernel_size_points)
# normalize the kernel
kernel = kernel / kernel.sum()
# Create a filtered signal by convolving the kernel with the original data
filtered = np.convolve(kernel, data, mode='valid')
```

```
# Display the result
fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(10, 3))
ax1.plot(np.arange(len(kernel))/sample_rate, kernel, '.-')
ax1.set_title("Kernel (1/50 sec wide)")
ax1.grid(alpha=.5, ls='--')
ax2.plot(np.arange(len(data))/sample_rate, data)
ax2.set_title("Original Signal")
ax2.margins(0, .1)
ax2.grid(alpha=.5, ls='--')
ax3.plot(np.arange(len(filtered))/sample_rate, filtered)
ax3.set_title("Convolved Signal")
ax3.margins(0, .1)
ax3.grid(alpha=.5, ls='--')
plt.tight_layout()
plt.show()
```

Azure Pipelines makes it easy to run tests in the cloud, but I found that a new React projects made with `create-react-app`

fail to properly test in the cloud using the simple `npm test`

command. Attempting this would display `No tests found related to files changed since last commit`

but hang forever.

I solved this problem and got my React app to test properly in the cloud by adding `-- --watchAll=false`

after `npm test`

. This is my final `azure-pipelines.yml`

file:

```
trigger:
- master
pool:
vmImage: "ubuntu-latest"
steps:
- task: NodeTool@0
inputs:
versionSpec: "10.x"
displayName: "Install Node.js"
- script: npm install
displayName: "Install NPM"
- script: npm run build
displayName: "Build"
- script: npm test -- --watchAll=false
displayName: "Test"
```

A working React app that tests properly with Azure Pipelines is GitHub.com/swharden/AliCalc

**This week my website was removed from the Wayback Machine.** The Wayback Machine is an impressive website that lets you view what a website looked like years ago. As part of Archive.org Internet Archive, this website is truly impressive and holds entertainingly-old versions of most webpages. Just look at Amazon.com in the year 2000 for a good laugh.

**I started this blog as a child twenty years ago** and after seeing what the Wayback Machine pulled-up I realized that it may be best that the thoughts I had as a child stay in the past. I have personal copies of all my old blog posts, but with the wisdom of age and hindsight I’d much prefer that that material stay off the internet. Luckily I was able to get my website removed from the Wayback Machine, and this post documents how I did it.

**For those of you wanting to do the same, this is how I did it:** I sent an email to `info@archive.org`

stating the following:

Please remove my website [MY URL] from the Wayback Machine.
[MY URL]/robots.txt has been updated to indicate I do not wish
this website to be archived.

https://lookup.icann.org/ shows that [MY URL] points to
[HOSTING COMPANY] nameservers, and I have attached a recent
invoice from [HOSTING COMPANY] as evidence that I own this domain.

If additional evidence or action is required (e.g., DMCA takedown
notice) please let me know.

Thank you!

Scott

**I’m not sure if editing **`robots.txt`

was necessary, but I felt it gave credence to the fact that I had control over the content of this domain. That file contains the following text. In the past I read this was all it took to get your website de-listed from the wayback machine, but I added this same file to another domain name of mine and it has not been de-listed.

```
User-agent: archive.org_bot
Disallow: /
User-agent: ia_archiver
Disallow: /
```

**I attached an invoice** from the present year showing a credit card payment to my hosting company for the domain as a PDF. Interestingly I did not have to show a history of domain ownership. I downloaded the invoice from my hosting company’s billing page that day, and it displays my home address but not my email address.

**Six days later, my site was removed.** This is the email I received:

FROM: Office Manager (Internet Archive)

Hello,

The following has now been submitted for exclusion from the
Wayback Machine at web.archive.org: [MY SITE]

Please allow up to a day for the automated portions of the process
to run their course and for the changes to take effect.

– The Internet Archive Team

**I reviewed a lot of websites** before reaching my strategy. I was surprised to see some people using issuing DMCA takedown notices notices to Archive.org, and was happy to find this was not required in my case. Here are some of the resources I found helpful:

**⚠️ WARNING:** This may not be permanent. I’m not sure what will happen if I lose my domain name (and robots.txt file) in the future. It is possible that my site is still being archived, while not being available on the wayback machine, and that some time in the future my site will be re-listed.

**If you have updated information** send me an email so I can update this page! In the mean time, I hope this information will be useful for others interested in curating their historical online presence.