Bilinear transformation method in Signal Processing - Time & Space Complexity

Time Complexity: Bilinear transformation method

O(n^2)

Understanding Time Complexity

We want to understand how the time needed to perform the bilinear transformation changes as the input size grows.

Specifically, how does the processing time scale when converting analog filter coefficients to digital ones?

Scenario Under Consideration

Analyze the time complexity of the following bilinear transformation code snippet.

// Given analog filter coefficients b, a and sampling frequency fs
function bilinearTransform(b, a, fs) {
  let degree = a.length - 1;
  let T = 1 / fs;
  let digitalB = new Array(degree + 1).fill(0);
  let digitalA = new Array(degree + 1).fill(0);

  for (let i = 0; i <= degree; i++) {
    for (let j = 0; j <= i; j++) {
      digitalB[i] += b[j] * combination(i, j) * Math.pow(2 / T, i - j);
      digitalA[i] += a[j] * combination(i, j) * Math.pow(2 / T, i - j);
    }
  }
  return { digitalB, digitalA };
}

// Helper function for combinations
function combination(n, k) {
  if (k === 0 || k === n) return 1;
  return combination(n - 1, k - 1) + combination(n - 1, k);
}

This code converts analog filter coefficients to digital ones using the bilinear transform method by calculating weighted sums with combinations.

Identify Repeating Operations

Look at the loops and repeated calculations:

• Primary operation: Nested loops over filter order (degree) for coefficient calculation.
• How many times: Outer loop runs (degree + 1) times, inner loop runs up to (degree + 1) times for each outer loop.
• Combination function: Called many times recursively inside loops.

How Execution Grows With Input

As the filter order (degree) increases, the number of operations grows roughly with the square of the degree because of nested loops.

Input Size (degree)	Approx. Operations
10	~100
100	~10,000
1000	~1,000,000

Pattern observation: Doubling the filter order roughly quadruples the number of operations.

Final Time Complexity

Time Complexity: O(n^2)

This means the time to compute the digital coefficients grows roughly with the square of the filter order.

Common Mistake

[X] Wrong: "The bilinear transform runs in linear time because it just loops once over coefficients."

[OK] Correct: There are nested loops and recursive calls inside, so the work grows faster than just one loop.

Interview Connect

Understanding how nested loops affect time helps you explain performance clearly and shows you can analyze real signal processing code efficiently.

Self-Check

"What if we replaced the recursive combination function with a precomputed lookup table? How would the time complexity change?"