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// signal processing project

Delay and tone, by hand

Finding an echo's delay by cross-correlation and removing one tone from a recording with a filter designed from first principles.

Two digital signal processing exercises in MATLAB and GNU Octave for the EENG527 course. Part A builds a decaying, noisy copy of a sine pulse and recovers its delay from the cross-correlation peak. Part B finds the dominant tone in a seven-second recording, designs a bandstop FIR filter with the window method and filters it out, writing the result back as audio.

source
// wave19/22
year
2026
type
signal processing project
group
systems
stars
0

//in plain words

Two small experiments in how computers handle signals. The first measures how late a noisy, fading echo of a beep arrives, by sliding the original over it until they line up. The second takes a recording with an annoying whistle in it, finds the whistle's pitch, and designs a filter that removes that one tone while leaving the rest of the sound alone.

for students meeting correlation and filter design, and anyone who wants the arithmetic behind a notch filter

the problem

//01the problem

A course on digital signal processing is full of formulas that only mean something once you have watched them work on a real signal. The assignment was to implement two of them without a toolbox doing the thinking: estimate a delay in noise, and remove a tone with a filter you sized yourself.

how it works

//02how it works

  1. 01

    Delay from the correlation peak

    A truncated sine pulse is copied into a longer signal at a chosen delay, scaled by an exponential decay and buried in Gaussian noise. Cross-correlating the noisy copy with the original gives a peak at the lag where they line up, and dividing by the sample rate turns it into seconds. The script asks for every parameter so the noise and decay can be pushed until the peak is lost.

  2. 02

    Find the tone before removing it

    The recording is made mono and normalised, and a zero-padded FFT finds the strongest component between 50 Hz and 8 kHz. A test tone at that frequency can be added so the whistle is easy to hear before and after.

  3. 03

    Size the filter from the specification

    The window is chosen from the required stopband attenuation: rectangular to 21 dB, Hann to 44, Hamming above. The length comes from the narrower transition band, rounded to odd so the centre tap is an integer and the phase is linear. The ideal bandstop is a delta minus the difference of two ideal lowpass responses, then multiplied by the window.

  4. 04

    Check the result in both domains

    The filtered audio is written next to the input, and figures show the signal and its spectrum before and after, the ideal and windowed impulse responses and the magnitude response in dB. After filtering, the band around the tone sits about 22 dB below the strongest remaining component.

measured

//03measured

0
samples of delay error
100 samples expected, 100 found, at 1 khz with noise on a decaying pulse
1500.01 Hz
tone detected
fft peak of a 7 s recording; the stopband is 1470 to 1530 hz
2,521
filter taps
hann window for a 40 db target, 70 hz transition bands at 44.1 khz

decisions

//04decisions

  1. 01

    chose the window method

    over an equiripple design

    The window method is the one the course teaches and every number in it can be derived by hand: the window from the attenuation, the length from the transition width. Equiripple designs are shorter but opaque.

  2. 02

    chose an odd filter length

    over the computed even length

    An odd length puts the centre tap on an integer sample, so the filter has exact linear phase and the delay it adds is a whole number of samples.

  3. 03

    chose random noise on every run

    over a fixed seed

    The point of part A is to see the peak survive noise. Each run draws fresh noise, so the plots change while the delay estimate should not.

try it

//05try it

  1. //install octave with the signal package

    brew install octave && octave --eval "pkg load signal; disp('signal ok')"

    fedora: dnf install octave octave-signal; ubuntu: apt install octave octave-signal

  2. //estimate a delay

    cd PartA && octave --persist PartA.m

    asks for eight values; the readme lists the example inputs

  3. //remove the tone

    cd PartB && octave --persist PartB.m

    writes audio_v1-filtered.wav next to the input

highlights

//06highlights

  • Cross-correlation delay estimate exact to the sample on a decaying, noisy pulse
  • Bandstop FIR designed by the window method from the attenuation spec
  • Runs in GNU Octave with the signal package; MATLAB needs two edits
  • Course project for EENG527, Digital Signal Processing

built with

  • MATLAB
  • GNU Octave
  • FFT
  • FIR filters

on githubEENG527-Project

stars
0
forks
0
language
MATLAB
last push
oct 9, 2026

printings

//07printings

  1. First commit

    Both parts, written in a day.

  2. README with the design rules

    Setup for three platforms, the window table, the length formula and the verified results.

questions

//08questions

Why a Hann window?
The specification asks for 40 dB of stopband attenuation. That falls between the 21 dB a rectangular window gives and the 44 dB Hann reaches, so Hann is the shortest window that meets it.
Why 2,521 taps?
The transition bands are 70 Hz wide at a 44.1 kHz sample rate. The Hann rule of eight pi over the transition width gives 2,520, plus one to make the length odd.
Does it run in MATLAB?
Yes, with the Signal Processing Toolbox and two edits the README lists: drop the package load line and replace one Octave-only function.

related sheets

//09related sheets

//10contact

Say hello. I read everything.

Email is the fastest way to reach me. For anything that needs a conversation, book a call.

book a callsave contact

to

name
Khaled Saeed
email
contact@khaledsaeed.tech
country
Lebanon