Fourier Domain Analysis

Prince E. Adjei

Kwame Nkrumah University of Science and Technology

Topic: Frequency Domain Analysis Module 1: Signal Analysis

Biosignal Processes And Analysis (BME 366)

Learning Objectives

(1). Explain the relationship between the time and frequency domains

(2). Differentiate between the Fourier Series, the Fourier Transform,

and the Discrete Fourier Transform (DFT), and apply the Fast Fourier

Transform (FFT) for efficient spectral analysis of physiological signals.

(3). Evaluate the impact of windowing and spectral leakage in

practical frequency analysis, and compare spectral estimation

methods.

(4). Apply the Fourier Transform to identify frequency components in

bio-signals.

Frequency-Domain Analysis

Topics:

(1). Frequency Domain Foundations

(2). Fourier Analysis Techniques

(3). Spectral Estimation and Windowing

(4). Frequency Characteristics of Physiological Signals

(5). Model-Based Signal Analysis

Review

•Time domain signal analysis involves examining how the signal

varies over time.

•Time-domain features are numerical characteristics derived

from the signal’s amplitude variation over time.

•They describe the shape, energy, and structure of the signal

without converting it into the frequency or other domains.

•These features are especially useful in biomedical signal

processing (like ECG, EMG, EEG), where signals are often

analyzed in real-time or on resource-constrained devices.

Frequency Domain Analysis

•Frequency domain analysis examines how a signal’s energy or power

is distributed across different frequency components.

•Unlike time domain analysis, it focuses on what frequencies make up

the signal rather than how the signal changes over time.

•This approach helps reveal underlying periods and oscillations within

the signal that may not be obvious in the time domain.

•Different physiological processes correspond to distinct frequency

bands, which can be identified using frequency domain analysis.

Fourier Series

•The Fourier Series is used to represent periodic signals (signals that repeat

exactly over time).

•Decomposes a periodic signal into a sum of discrete sinusoidal frequencies

(harmonics).

•Frequencies are integer multiples of the fundamental frequency.

•Example: A pure tone or a repetitive heartbeat pattern.

•Mathematically :

•where T is the period of the function, 𝒂𝒐is the average value (DC component)

of the function over one period, 𝒂𝒏and 𝒃𝒏are the Fourier coefficients which

represent the amplitudes of cosine and sine at each harmonic n

Fourier Transform

•The Fourier Transform is used to analyze aperiodic or non-repeating

signals

•Converts a time-domain signal into a continuous spectrum of

frequencies

•Useful for signals like transient events or signals with no strict

periodicity.

•Example: An ECG recording, which is quasi-periodic, mostly

repetitive but with variations.

Continuous Fourier Transform

•The continuous fourier transform, transforms a continuous-time

signal f(t) into its continuous frequency spectrum F(ω).

•It shows how much of each frequency is present in the signal.

•Mathematically:

Where

•F(ω): frequency domain representation

•f(t): original continuous signal in the time domain

•ω: angular frequency (radians/second)

•𝒆−𝒋𝝎𝒕: complex sinusoid basis functions

Discrete Fourier Transform

•The discrete fourier transform transforms a discrete sequence x[n]

(digital samples) into discrete frequency components X[k].

•Mathematically:

Where

•x[n]: discrete signal samples (time domain)

•X[k]: discrete frequency components

•N: total number of samples

•e−jN2πkn: discrete complex sinusoids

Inverse Fourier Transform

•The inverse fourier transform reconstructs the original signal from its

frequency components.

•Continuous inverse:

•Discrete inverse:

•f(t),x[n]: reconstructed signals in time domain

•F(ω), X[k]: frequency domain signals

•The exponential sign flips to positive for reconstruction.

Discrete-Time Fourier Transform (DTFT)

••The Discrete-Time Fourier Transform (DTFT) is a mathematical tool

that transforms a discrete-time signal from the time domain to the

frequency domain.

••In an LTI system, the output y[n] is the convolution of input x[n] with

the impulse response h[n]:

••Step 1: Start from time-domain convolution

•Step 2: Take the z-transform of both sides

•Let:

•Then

Because convolution in time ⟷multiplication in the z-domain

Discrete-Time Fourier Transform (DTFT)

•Step 3: Evaluate on the unit circle

•DTFT is the z-transform evaluated on the unit circle

•So, applying this to the output

Discrete-Time Fourier Transform (DTFT)

•The DFT converts N time-domain samples into N frequency bins:

•x[n]: discrete-time signal (length N)

•X[k]: complex spectrum at bin k

•e-term: rotating phasor (basis function)

Discrete Fourier Transform (DFT)

•Bin spacing is the difference in frequency between two adjacent

DFT output points (bins).

•Mathematically:

Where:

•fs: sampling frequency (Hz)

•N: number of samples

•Δf: bin spacing (frequency resolution)

Discrete Fourier Transform (DFT)

Interpretation:

•Each index k in the DFT output corresponds to a frequency:

Smaller Δf (better frequency resolution) is achieved by:

•Increasing N (more samples)

•Keeping fs fixed

•Example If fs = 1 kHz and N = 1000 →bin width = 1 Hz →X[5]

represents the 5 Hz component

Discrete Fourier Transform (DFT)

•The Fast Fourier Transform(FFT) is a fast algorithm to compute

the Discrete Fourier Transform(DFT).

•DFT is too slow for real-time signal analysis.

•FFT speeds it up from: O(𝑵𝟐) to O(NlogN)

Fast Fourier Transform

•Most common FFT : Radix-2 Decimation-in-Time (DIT)

•Breaks the signal of length N into smaller and smaller parts(halves).

•Uses a basic unit called the butterfly.

•Butterfly operation:

•Where is a twiddle factor.

Fast Fourier Transform

Questions

1. What if frequency domain

analysis?

2. What is the key difference

between the Fourier series

and transform?

3. What is the difference

between DFT, FFT, and DTFT?

4. Why is FFT preferred over DFT

in practice?

Spectral Leakage

•Spectral leakage happens when a signal’s frequency doesn’t align

with a DFT bin.

The DFT assumes:

•The signal is periodic within the window.

•It fits perfectly into a set of discrete frequency bins.

•When it doesn’t, the energy spreads into adjacent bins

Spectral Leakage

Why does this happen?

•When analyzing a finite-length signal. That is equivalent to

windowing it in time.

•Windowing multiplies the signal by a rectangular function, which

in the frequency domain becomes a sinc function.

•This sinc smearing causes the main frequency to "leak" into

neighboring bins.

Spectral Leakage

•A pure sine wave at a bin frequency (e.g., exactly 50.0 Hz)

produces a clean, single-bin peak.

•A sine wave at 50.3 Hz, which is not aligned with a bin, causes

energy to leak into nearby bins.

Check:

•If the frequency/bin spacing ≠integer, expect leakage.

Formula:

•If the bin index is not an integer, spectral leakage will occur.

Windowing

•Windowing involves multiplying the signal with a window function

before performing the FFT.

•The goal is to minimize discontinuities at the edges of the FFT

segment, which cause spectral leakage.

••This window shapes the frequency response.

•Think of windowing as a filter: it controls how energy is spread

across frequencies in the FFT

Windowing (Rectangular vs Hann)

-3 dB Main-Lobe Width Trade-off

•The -3 dB point is where the main lobe’s power drops to half.

Trade-off:

•Rectangular window gives better resolution (narrow main lobe),

but suffers from more leakage (high side lobes).

•Hann window gives less leakage (low side lobes), but poorer

resolution (wider main lobe).

Windowing (Rectangular vs Hann)

Power Spectral Density

•Power Spectral Density (PSD) is a statistical measure used to

describe how the power of a signal or time series is distributed

across different frequency components.

•It quantifies the power present in the signal as a function of

frequency.

•The units of PSD are typically watts per hertz (W/Hz) or decibels

(dB)

Power Spectral Density

•The PSD can be represented mathematically as:

•Where:

•S(f) is the power spectral density at frequency ff.

•x(t) is the time-domain signal.

•j is the imaginary unit.

Periodogram

••Aperiodogram is an estimate of the power spectral density (PSD)

of a signal, based on the squared magnitude of its Discrete Fourier

Transform (DFT).

•Mathematically:

Where

•x[n] is the signal

•Nis the number of samples

•𝑭𝒌is the frequency bin

•P(𝒇𝒌)is the power at frequency 𝒇𝒌

Averaged PSD (Welch Method)

•An improved version of the periodogram that reduces variance by

averaging several periodograms.

Steps in Welch’s Method:

1. Segmenting

•The signal is split into overlapping segments (e.g., 50% overlap).

•Each segment is assumed to be locally stationary.

2. Windowing

•A window function (e.g., Hamming) is applied to each segment to

reduce spectral leakage from sharp edges.

Averaged PSD (Welch Method)

3. FFT & Power:

•FFT is computed on each windowed segment.

•The squared magnitude of the Fourier transform of each

windowed segment yields its corresponding power spectrum.

4. Averaging:

•The power spectra of all segments are averaged to get a reliable

PSD estimate.

Periodogram vs averaged PSD ECG demo

Questions

1. What causes spectral leakage

during FFT analysis?

2. How does windowing reduce

spectral leakage?

3. Why is the Welch’s method

preferred over the basic

periodogram for estimating

PSD?

Physiological band examples

ECG (Electrocardiogram): < 40 Hz

•Most of the energy of the ECG signal lies below 40 Hz.

•High-pass filters around 0.5 Hz and low-pass filters at 40 Hz are often

used for noise removal (e.g., baseline wander and muscle artifacts).

EMG (Electromyogram): 20 –450 Hz

•Captures muscle activity.

•Often filtered between 20–450 Hz to eliminate motion artifacts (low

frequency) and electrical noise (high frequency).

•Diagnostic studies may extend up to 1000 Hz.

Physiological band examples

EEG (Electroencephalogram):

•EEG is divided into specific rhythmic bands:

•A signal can be modeled as a linear combination of its past values

plus random noise:

Autoregressive(AR) Modelling

•Models signals with hidden structure (e.g., EEG, ECG, EMG)

•Compact representation (low parameter count)

•Captures oscillatory or predictable behavior (like alpha rhythms

in EEG)

Example Applications

•Detecting changes in EEG frequency content

•Smoothing EMG envelopes

•Feature extraction for classification

Why Use AR Models in Biosignals?

•Efficiently computes AR coefficients from the signal’s

autocorrelation.

•It is much faster than solving normal equations directly.

•Used to fit the AR model:

Levinson-Durbin Algorithm

•Step by Step overview :

1.Initialize

2.Recursion for k = 2 to p

•Compute reflection coefficient:

Levinson-Durbin Algorithm

3. Update AR coefficients:

4. Update prediction error:

Levinson-Durbin Algorithm

Pseudo Code :

Levinson-Durbin Algorithm

AR ECG Case Study

Time-Domain ECG-Like Signal (Top Plot)

•This shows a simulated ECG signal over 5 seconds, mimicking the

key features of a real ECG. It's a clean signal with a bit of added

noise to reflect typical biosignal conditions.

Periodogram (Middle Plot)

•Here we use a Welch periodogram, a non-parametric spectral

estimation method based on the FFT.

•It provides a detailed frequency spectrum, but it’s sensitive to noise

and shows spectral leakage, especially when data segments are short

or noisy.

AR ECG Case Study

AR Models vs Periodogram (Bottom Plot)

•This graph compares PSDs from AR models of orders 8, 10,and 12

with the periodogram.

•The AR models give smoother spectra that still capture key peaks

(like those in the 10–25 Hz band, typical of QRS energy).

•As order increases, the model gets more flexible ,but if it's too high,

it may overfit.

AR ECG Case Study

Periodogram vs AR model ECG

ARMA extension

•This time-domain signal is

generated by filtering white

noise with a 2-pole, 2-zero

ARMA filter, tuned to

amplify ~10 Hz oscillations —

representing the alpha

rhythm seen in EEG during

relaxed wakefulness.

•The result is a smooth, rhythmic waveform centered around 10 Hz,

mimicking real occipital EEG activity with minimal external artifacts.

ARMA extension

•This shows the gain (in dB)

across frequencies for the

ARMA filter, with a distinct

peak at ~10 Hz, confirming

the filter is tuned to

enhance alpha-band

activity.

•Frequencies outside the alpha band are attenuated, illustrating how

filter design shapes signal characteristics in the frequency domain.

ARMA extension

•Poles (red ×)lie close to the unit

circle at angles corresponding

to ±10 Hz, indicating strong

resonance at that frequency.

•Zeros (green ○)control the spectral shape, and their position helps

suppress or enhance specific frequency bands, showcasing spectral

shaping via zero placement.

ARMA extension

•This frequency-domain

representation of the filtered

signal shows a dominant

spectral peak at 10 Hz,

validating the filter’s role in

isolating the alpha rhythm.

•The clean, sharp peak highlights the effectiveness of ARMA modeling

in capturing oscillatory features even from random input.

Summary

•Frequency domain analysis examines how a signal’s energy or power is distributed across

different frequency components.

•Fourier Series: For periodic signals, it decomposes into sine/cosine components.

•Fourier Transform: For aperiodic signals, it gives a continuous frequency spectrum.

•DFT: Discrete samples → discrete frequencies; used in DSP.

•FFT: Fast algorithm to compute the DFT efficiently (O(N log N)).

•Spectral leakage occurs when a signal’s frequency doesn’t align with DFT bins.

Summary

•Windowing reduces spectral leakage by smoothing signal edges.

•PSD quantifies how signal power is distributed across frequencies (W/Hz or

dB).

•Periodogram vs Welch Method

•Periodogram: PSD from squared DFT magnitude; simple but noisy.

•Welch’s Method: Reduces variance by segmenting, windowing, and averaging

PSD estimates across segments.

References

•Rangayyan, R. M. (2015). Biomedical signal analysis: A case-study approach

(2nd ed.). IEEE Press Series in Biomedical Engineering. Wiley–IEEE Press.

ISBN: 978-0-470-01139-6.

•Palaniappan, R. (2011). Biological signal analysis. University of Essex