By Aaron Shippey | Aug 27, 2026
Today I dove back into DSP for a moment. I wanted to keep it simple and focus on the fundamentals.
A core concept in digital audio is representing a continuous-time signal as a discrete sequence of samples. The project I worked on today was a simple implementation of an exponential decay function. I wanted to understand how something as basic as an envelope can be represented mathematically and then implemented digitally.
The first step was to establish Euler’s number:
The exponential decay function I’m using is:
where k is the decay-rate constant and t is time.
For k, I picked an arbitrary value of 5. At this point, I hadn’t calculated t yet.
At first, I thought I could just create a list to represent our discrete sequence. But a loop made more sense because I didn’t just want to create a list of numbers. I wanted to understand what was actually happening as we moved through the sequence and calculated the envelope.
So I created a variable called n to represent the sample index.
The goal was simple: iterate through the sample index so that I could calculate the corresponding point in time and then calculate the exponential decay at that point.
Because the first index in a discrete sequence is 0, the first calculation occurs at:
With a sample rate of 10 samples per second (fs = 10), we calculate time from the sample index:
So for the first sample:
We then evaluate the exponential decay function at that time:
The first envelope value is 1.
On the next iteration, n is incremented from 0 to 1.
Now:
Next, we’re evaluating the same exponential decay function at 0.1 seconds:
Which means our first two envelope values are approximately:
This was where the distinction between the sample index, time, and envelope value became much clearer to me.
n tells us which sample we’re at. The sample rate tells us how many samples occur per second. From those two things, we can determine the time represented by that sample:
Then we use that time in the exponential decay function to calculate the envelope value.
The basic chain is:
This exploratory work is central to how I’m building the technical foundation for PluginProof. I don’t build DSP or develop audio plugins myself; my focus is on testing software and providing specialized technical QA for the developers who do.
The goal here isn’t to become the engineer writing signal processing algorithms from scratch. Instead, it’s about understanding the underlying technology deeply enough to ask sharper questions, design more rigorous test suites, spot unexpected software behavior, and deliver clear, actionable QA documentation.
Working through this program made me realize that there’s quite a bit more going on underneath what initially looks like a simple sequence of numbers.