Applying filters (lfilter, sosfilt) in SciPy - Step-by-Step Execution

Concept Flow - Applying filters (lfilter, sosfilt)

Start: Input Signal

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Choose Filter Type

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lfilter

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Apply filter coefficients

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Filtered Output Signal

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End

The input signal is processed by choosing either lfilter or sosfilt, applying filter coefficients or second-order sections, producing the filtered output.

Execution Sample

SciPy

from scipy.signal import lfilter, sosfilt, butter
b, a = butter(2, 0.3)
x = [1, 2, 3, 4, 5]
y = lfilter(b, a, x)

This code creates a Butterworth filter and applies it to a simple signal using lfilter.

Execution Table

Step	Input x[n]	Filter Coefficients b	Filter Coefficients a	Calculation	Output y[n]
1	1	[0.2066, 0.4131, 0.2066]	[1.0, -0.3695, 0.1958]	y[0] = b0*x[0]	0.2066
2	2	[0.2066, 0.4131, 0.2066]	[1.0, -0.3695, 0.1958]	y[1] = b0*x[1] + b1*x[0] - a1*y[0]	0.9026
3	3	[0.2066, 0.4131, 0.2066]	[1.0, -0.3695, 0.1958]	y[2] = b0*x[2] + b1*x[1] + b2*x[0] - a1*y[1] - a2*y[0]	1.9457
4	4	[0.2066, 0.4131, 0.2066]	[1.0, -0.3695, 0.1958]	y[3] = b0*x[3] + b1*x[2] + b2*x[1] - a1*y[2] - a2*y[1]	3.0923
5	5	[0.2066, 0.4131, 0.2066]	[1.0, -0.3695, 0.1958]	y[4] = b0*x[4] + b1*x[3] + b2*x[2] - a1*y[3] - a2*y[2]	4.1721
6	-	-	-	-	End of signal, filtering complete

💡 Reached end of input signal x, no more samples to filter.

Variable Tracker

Variable	Start	After 1	After 2	After 3	After 4	After 5	Final
x[n]	[1, 2, 3, 4, 5]	1	2	3	4	5	-
y[n]	[0, 0, 0, 0, 0]	0.2066	0.9026	1.9457	3.0923	4.1721	4.1721

Key Moments - 3 Insights

Why does the output y[n] depend on previous outputs y[n-1], y[n-2]?

Because the filter is recursive (IIR), each output uses past outputs as feedback, shown in execution_table rows 2-5 where a1 and a2 coefficients multiply previous y values.

What is the difference between lfilter and sosfilt in applying filters?

lfilter uses direct filter coefficients (b, a), while sosfilt uses second-order sections for better numerical stability, especially for higher order filters.

Why do we need to use butter() before lfilter or sosfilt?

butter() designs the filter and returns coefficients needed by lfilter or sosfilt to apply the filter to the signal.

Visual Quiz - 3 Questions

Look at the execution_table, what is the output y[3] after filtering the input x[3]?

A 3.0923

B 1.9457

C 0.9026

D 4.1721

💡 Hint

Check the row where Step is 4 in the execution_table for y[3].

At which step does the filtering process stop according to the execution_table?

A Step 5

B Step 4

C Step 6

D Step 3

💡 Hint

Look for the row with 'End of signal' in the Calculation column.

If the input signal x had one more sample, how would the variable_tracker change?

A It would remove the last column.

B It would add one more column for After 6 with new x and y values.

C It would keep the same columns but change y values.

D It would reset all values to zero.

💡 Hint

Variable tracker columns represent steps after each input sample is processed.

Concept Snapshot

Applying filters with scipy.signal:
- Use butter() to design filter coefficients (b, a) or sos.
- Use lfilter(b, a, x) for direct filtering.
- Use sosfilt(sos, x) for stable filtering with second-order sections.
- Output depends on current and past inputs and outputs (IIR filters).
- Filtering stops when input signal ends.

Full Transcript

This visual execution shows how scipy.signal applies filters using lfilter and sosfilt. Starting with an input signal, we design filter coefficients using butter(). Then lfilter applies these coefficients step-by-step to produce the filtered output. Each output sample depends on current and previous inputs and previous outputs, demonstrating recursive filtering. The execution table traces each calculation and output value. The variable tracker shows how input and output values change after each step. Key moments clarify why outputs depend on past outputs and the difference between lfilter and sosfilt. The quiz tests understanding of output values, stopping conditions, and variable tracking. This helps beginners see filtering as a stepwise process, not just a black box.