"Revealing the Hidden Frequencies : Exploring the Discrete Fourier Transform"

Revealing the Hidden Frequencies: Exploring the Discrete Fourier Transform (DFT)

Author: Anita S Lokare
Domain: Digital Signal Processing
Mentors: Dr. Kiran TALELE & Dr. Reena Kumbhare

Beginning the Journey into Frequencies

Consider an audio clip that contains an important voice message but is corrupted by unwanted high-frequency noise. In the time domain, this interference appears as unpredictable spikes and irregular oscillations. Although distortion is visible, separating useful sound from noise becomes difficult.

This blog explains how filtering affects both the time domain and the frequency domain by comparing an audio signal before and after filtering.

Key Parameters

• Sampling Frequency = 44.1 kHz
• Filter Type = FIR Low Pass Filter
• Cutoff / Stopband Frequency = 6000 Hz
• Filter Order = 100 taps
• Passband Ripple = 1 dB
• Stopband Attenuation = 60 dB

Statistical Interpretation

• High magnitude → strong frequency present
• Low magnitude → weak / noise
• DFT helps compute Power Spectral Density (PSD)

Time Domain

The time domain shows how a signal varies over time (amplitude vs time). But from this view alone, the exact frequencies cannot be identified.

Example: A mixed audio signal looks like one waveform but contains multiple sine waves of different frequencies.

Frequency Domain

The frequency domain shows how much of each frequency exists inside the signal. By applying the DFT,we convert the signal from the time domain to the frequency domain.

Frequency peaks represent hidden sinusoidal components.

Insights from the DFT

• The original audio contains high-frequency fluctuations.
• After filtering, high-frequency components beyond cutoff are removed.
• The filtered signal becomes smoother in the time domain.
• The frequency domain spectrum clearly shows reduced high-frequency magnitude.

Scope for Enhancement

• Use a higher-order filter for sharper cutoff.
• Apply Hamming / Hanning / Blackman windows to reduce spectral leakage.
• Increase FFT size or use zero-padding for better resolution.

Final Analysis

DFT converts a signal from the time domain to the frequency domain. The time domain only shows amplitude variation, not hidden frequencies. The frequency domain clearly reveals frequency strength and distribution. DFT is essential in audio processing, communication systems, radar, and biomedical signals.

The frequency domain clearly reveals frequency strength and distribution. DFT is essential in audio processing, communication systems, radar, and biomedical signals.

References

• Oppenheim & Schafer – Discrete-Time Signal Processing
• Proakis & Manolakis – Digital Signal Processing
• Lyons – Understanding Digital Signal Processing
• Cooley & Tukey – Algorithm for Complex Fourier Series