IMPULSE_RESPONSE
The impulse response measures how a linear system reacts when its input is an ideal unit impulse, represented mathematically by the Dirac delta function \delta(t).
For a transfer function G(s), the impulse response is the inverse Laplace transform of that transfer function:
g(t) = \mathcal{L}^{-1}\{G(s)\}
Because the impulse response characterizes the full time-domain dynamics of an LTI system, it is a standard tool for understanding transient behavior. This wrapper returns a 2D array whose first row is the time vector and whose second row is the response values.
Excel Usage
=IMPULSE_RESPONSE(sysdata, timepts)sysdata(list[list], required): The system transfer function, provided as numerator and denominator coefficients in consecutive rows.timepts(list[list], optional, default: null): Optional time vector or simulation duration. If left blank, it is auto-computed based on the system’s dynamics.
Returns (list[list]): A 2D array where the first row contains the time vector values and the second row contains the system impulse response output.
Example 1: First order impulse response
Inputs:
| sysdata | timepts | |
|---|---|---|
| 1 | 0 | 1 |
| 1 |
Excel formula:
=IMPULSE_RESPONSE({1;1,1}, {0,1})Expected output:
| Result | |
|---|---|
| 0 | 1 |
| 1 | 0.367879 |
Example 2: Lightly damped impulse
Inputs:
| sysdata | timepts | |
|---|---|---|
| 100 | 0 | 0.1 |
| 1 |
Excel formula:
=IMPULSE_RESPONSE({100;1,2,100}, {0,0.1})Expected output:
| Result | |
|---|---|
| 0 | 0.1 |
| 0 | 7.62758 |
Python Code
Show Code
import control as ct
import numpy as np
def impulse_response(sysdata, timepts=None):
"""
Compute the impulse response for a linear system.
See: https://python-control.readthedocs.io/en/latest/generated/control.impulse_response.html
This example function is provided as-is without any representation of accuracy.
Args:
sysdata (list[list]): The system transfer function, provided as numerator and denominator coefficients in consecutive rows.
timepts (list[list], optional): Optional time vector or simulation duration. If left blank, it is auto-computed based on the system's dynamics. Default is None.
Returns:
list[list]: A 2D array where the first row contains the time vector values and the second row contains the system impulse response output.
"""
try:
def to_row_list(values):
arr = np.asarray(values)
if arr.ndim == 0:
return [float(arr)]
return [float(value) for value in arr.reshape(-1)]
def flatten_to_1d(x):
if not isinstance(x, list):
return [float(x)] if str(x) != "" else []
flat = []
for row in x:
if isinstance(row, list):
for val in row:
if str(val) != "" and val is not None:
flat.append(float(val))
elif row is not None and str(row) != "":
flat.append(float(row))
return flat
if not sysdata or len(sysdata) < 2:
return "Error: sysdata must contain numerator and denominator rows"
num = flatten_to_1d(sysdata[0])
den = flatten_to_1d(sysdata[1])
if not num or not den:
return "Error: Invalid numerator or denominator arrays"
sys = ct.tf(num, den)
t_in = flatten_to_1d(timepts) if timepts is not None else None
if t_in is not None and len(t_in) == 1:
t_in = float(t_in[0])
elif t_in and len(t_in) == 0:
t_in = None
resp = ct.impulse_response(sys, T=t_in)
t_list = to_row_list(resp.time)
y_list = to_row_list(np.squeeze(resp.outputs))
return [t_list, y_list]
except Exception as e:
return f"Error: {str(e)}"