Time–Frequency Transforms: STFT vs Wavelets

Prince E. Adjei

Kwame Nkrumah University of Science and Technology

Topic: Time–Frequency Transforms Module 4: Advanced Topics

Biosignal Processes And Analysis (BME 366)

Topics

(1). Time–Frequency Analysis Basics

(2). Short-Time Fourier Transform (STFT)

(3). Wavelet Transform

(4). Wavelet Bases

(5). Wavelet Application

Time–Frequency Transforms: STFT vs Wavelets

Learning Objectives

•Describe the need for time–frequency analysis in processing non-

stationary biomedical signals.

•Compare the Short-Time Fourier Transform (STFT) and Wavelet

Transform in terms of resolution and adaptability.

•Interpret wavelet basis functions such as Haar and Daubechies,

including their orthogonality and structure.

•Apply Discrete and Continuous Wavelet Transforms (DWT/CWT) to

analyze and process biomedical signals.

•Use wavelet-based methods for tasks such as denoising and systolic

peak detection in signals like PPG.

Review

Time domain shows how a signal changes over time.

•Useful for identifying peaks, durations, and intervals (e.g., R-peaks in ECG).

•Provides a direct and intuitive view of the signal’s shape.

•Ideal for detecting event timing and analyzing waveform patterns.

Frequency domain shows which frequencies are present in the signal and how

strong they are.

•Useful for uncovering hidden rhythms, periodic components, or noise.

•Achieved using tools like the Fourier Transform or Power Spectral Density

(PSD).

•Ideal for analyzing spectral content and understanding how energy is

distributed.

Time-Frequency Transforms

•Many biosignals like ECG, EEG, and PPG are non-stationary, meaning

their frequency content changes over time.

•Traditional Fourier Transform gives global frequency information but

loses all time localization.

•It tells you what frequencies exist, but not when they occur.

•In real-world biomedical signals, important events (e.g., arrhythmias,

muscle bursts) are transient ; they happen for a short duration and may

not be visible in a full-spectrum view.

Time-Frequency Transforms

Time–Frequency analysis solves this by combining both domains:

•It shows how frequency content evolves over time.

•Helps in identifying dynamic signal behavior, detecting artifacts, or

understanding physiological changes.

Common time–frequency tools include:

•Short-Time Fourier Transform (STFT) –uses sliding windows.

•Wavelet Transform –uses variable-resolution windows for better

adaptability.

Short-Time Fourier Transform (STFT)

•STFT analyzes a non-stationary signal by dividing it into short, fixed-length

windows, then applying the Fourier Transform to each window.

Strengths

•Provides both time and frequency information.

•Simple and widely used —easy to implement.

•Works well when signal changes are slow or moderate.

Limitations

•Has constant resolution:

•Fixed window size = fixed trade-off between time and frequency resolution.

•Anarrow window gives good time resolution but poor frequency resolution.

•Awide window gives good frequency resolution but poor time resolution.

•Not ideal for signals with both fast and slow changes, like many biosignals.

Continuous-Time STFT

•The Continuous-Time STFT is a mathematical way to look at how the

frequencies in acontinuous signal change over time.

•It slides a window over the signal and applies the Fourier Transform to just

that small part.

•This shows what frequencies are present at each time point.

•Mathematically:

•x(τ): your signal

•w(τ−t): the sliding window centered at time t

•𝒆−𝒋𝟐𝝅𝒇𝝉: the Fourier part (to analyze frequency f)

•The Discrete-Time STFT is the digital version of STFT, used for analyzing

signals on a computer; like an ECG recorded as a list of values.

•It takes short segments (windows) from a digital signal.

•For each segment, it computes the Discrete Fourier Transform (DFT).

•This shows what frequencies are present at each time step.

•x[m]: the discrete signal

•w[m−n]: the window centered at position n

•𝒆−𝒋𝟐𝝅𝒌𝒎/𝑵: computes the DFT at frequency bin k

•N: total number of frequency bins (FFT size)

Discrete-Time STFT

•The Wavelet Transform is a tool for analyzing signals whose properties

change over time but with one big advantage: variable-width Windows

•Unlike STFT (which uses a fixed-size window), wavelets use:

⚬Narrow windows for high-frequency components →better time

resolution.

⚬Wide windows for low-frequency components →better frequency

resolution.

•This gives adaptive resolution, making wavelets better for non-stationary

signals like ECG or EMG.

At each level, it separates the signal into:

•Approximation (A) –low-frequency components

•Detail (D) –high-frequency components

Wavelets

Step-by-Step Breakdown

1. Filtering

•The signal is passed through two filters:

⚬A low-pass filter (LPF) →keeps slow-changing parts (approximation).

⚬A high-pass filter (HPF) →keeps fast-changing parts (detail).

2. Downsampling

•After filtering, the outputs are downsampled by 2(keep every second

sample) to reduce redundancy.

So after one level, we get:

•A₁ (approximation at level 1) and D₁ (detail at level 1)

3. Recursive Decomposition

•The process is repeated on the approximation only:

⚬A₁ →LPF + HPF →A₂ and D₂ and so on…

Wavelets

1.Why is time–frequency analysis

important for non-stationary signals

like PPG or ECG?

2.What trade-off does the Short-Time

Fourier Transform (STFT) face when

selecting a window size?

3.How do Haar and Daubechies

wavelets differ in terms of

smoothness and signal

representation?

4.What is the main advantage of

wavelet transforms over STFT in

analyzing biosignals?

Questions

•Wavelet transforms use small “wave-like” functions (called mother

wavelets) to analyze signals at different scales. Two common families are

Haar and Daubechies.

1. Haar Wavelet

•Simplest wavelet: looks like a step function.

•It's like a square pulse ; sharp and localized in time.

•Great for capturing sudden changes, edges, or spikes.

•Fast and efficient, but not smooth —can introduce blocky artifacts.

2. Daubechies

•Smoother and more complex than Haar —better at capturing gradual

changes.

•db2is the simplest, db4, db6, etc., give better frequency localization but are

longer.

Haar and Daubechies

Feature

Haar

Daubechies

Shape

Step

-like (square wave)

Smooth and oscillatory

Time resolution

Excellent (very localized)

Good

Frequency resolution

Poor (blocky)

Better (smoother filtering)

Use case

Quick detection of edges/spikes

Denoising, ECG features, smooth

signals

•Orthogonal wavelets form a complete, non-overlapping basis.

This means:

•No redundancy: each component (detail or approximation) is unique.

•Perfect reconstruction: you can fully recover the original signal from

its wavelet coefficients.

Both Haar and Daubechies wavelets are orthogonal, making them suitable

for:

•Lossless signal analysis

•Efficient storage

•Exact inverse transforms

Orthogonality

•The DWT breaks a signal into different frequency bands using wavelet filters.

•It separates the signal into two components at each level:

⚬Approximation (A) –low-frequency part

⚬Detail (D) –high-frequency part

•This process is repeated only on the approximation to get a multilevel

breakdown.

Multilevel Decomposition

•At Level 1, the signal is split into:

⚬A1 (approximation)

⚬D1(detail)

•At Level 2, the approximation A1 is split again: A2, D2

•This continues for multiple levels:

⚬Level 3 →A3,D3

⚬Level 4 →A4,D4 and so on.

Discrete Wavelet Transform (DWT)

•Example (4-level DWT):

•Each level gives finer time resolution for high frequencies (D components)

and better frequency resolution for low frequencies (A components).

Discrete Wavelet Transform (DWT)

•The CWT analyzes a signal by convolving it with scaled and shifted versions

of a wavelet function.

•Unlike DWT, CWT works over a continuous range of scales and positions,

giving smooth, high-resolution time–frequency analysis.

Morlet Wavelet

•A commonly used continuous wavelet transform (CWT)(sine wave multiplied

by a Gaussian).

•Good for analyzing oscillatory signals like PPG, EEG, EMG.

•Looks like a short wave packet —ideal for time–frequency analysis.

•Mathematical form:

•f0: central frequency

•Combines time localization (Gaussian) and frequency sensitivity (sine wave)

Continuous Wavelet Transform (CWT)

Scale to Frequency Conversion

•In CWT, you analyze the signal at different scales (a) —but we often want to

know the corresponding frequencies.

•f: actual frequency

•a: scale

•fc: center frequency of the wavelet

So:

•Small scale → high frequency

•Large scale → low frequency

Continuous Wavelet Transform (CWT)

1.How are Haar and Daubechies

wavelets different?

2.What does DWT do to a signal at

each level?

3.What does the Morlet wavelet help

with in CWT?

4.How is scale related to frequency

in CWT?

Questions

•PPG (Photoplethysmogram) signals often contain noise from:

⚬Motion artifacts

⚬Powerline interference (50/60 Hz)

⚬Baseline wander

•Wavelet denoising removes high-frequency noise while preserving sharp

systolic peaks.

Wavelet Denoising Steps (DWT-based):

1.Decompose signal using DWT into A + D coefficients.

2.Threshold the detail coefficients (D) to suppress noise.

3.Reconstruct the signal using the modified coefficients.

Denoising for PPG

•The systolic peak in a PPG signal has a sharp rising edge (steep up-

slope).

•The 1st level detail coefficients (D1) from a wavelet transform (like using

db4) capture high-frequency components, especially:

⚬Sudden transitions

⚬Sharp slopes

•The positive peak in D1 corresponds to the fast up-slope of the systolic

wave.

Systolic Detection via Wavelet D1 (Up-Slope Method)

Step-by-Step Detection Strategy

1.DWT Decomposition:

•Apply 1-level DWT to the PPG signal using a wavelet (e.g., db4).

•Get the D1 coefficients.

2. Identify Rising Edges:

•Look for positive peaks in D1 —they indicate fast increases in the signal.

•These often align with onsets of systolic waves.

3. Refine Detection:

•Cross-reference with the original signal to:

•Locate actual peaks

•Filter out false positives (e.g., from noise)

Systolic Detection via Wavelet D1 (Up-Slope Method)

Summary

Why Time–Frequency Analysis?

•Biosignals like PPG and ECG are non-stationary

•Require methods that show both time and frequency changes

STFT –Strengths & Limitations

•Uses a fixed-size window

•Trade-off:better time or frequency resolution, not both

•Good for signals with relatively constant frequencies

Wavelet Transform (DWT & CWT)

•Uses variable window sizes: short for high freq, long for low

•DWT:gives discrete levels (A1–A4, D1–D4)

•CWT with Morlet: continuous scales, allows scale-to-frequency

conversion.

Summary

Wavelet Types –Haar vs Daubechies

•Haar: simple, sharp edges

•Daubechies: smoother, better for real biosignals

•Both are orthogonal, useful for decomposition and reconstruction

Applications

•PPG Denoising: threshold high-frequency details

•Systolic Detection: use D1 to find sharp up-slopes in PPG

•Allows noise suppression and peak tracking in one framework