Fourier Transform Applications in Engineering | ECExplain

Visual

Concept overview

The Fourier Transform decomposes a signal from its time-domain representation into its constituent frequency components. For a continuous signal, the transform produces a complex spectrum where magnitude indicates the amplitude of each frequency and phase indicates its timing offset. The Discrete Fourier Transform (DFT) operates on sampled sequences, and the Fast Fourier Transform (FFT) algorithm computes the DFT in O(N log N) operations rather than O(N squared), making real-time spectral analysis practical. In engineering, the transform is the primary tool for understanding the frequency content of electrical signals, mechanical vibrations, and image data.

Real-world applications

Spectrum Analysis in Rohde and Schwarz FSW

Test and Measurement

The R&S FSW signal and spectrum analyser uses a swept FFT engine to display the power spectral density of RF signals from 2 Hz to 90 GHz. The IF signal chain digitises the downconverted signal at up to 10 GSPS, and the DSP computes overlapping FFT blocks to produce a continuously updated spectrum display used by RF engineers during transmitter conformance testing.

OFDM Modulation in 4G LTE

Wireless Communications

LTE uses Orthogonal Frequency Division Multiplexing where the IFFT at the transmitter maps data symbols onto orthogonal subcarriers separated by 15 kHz. The FFT at the receiver recovers the data from each subcarrier independently. The 3GPP standard defines resource blocks of 12 subcarriers, with FFT sizes of 128 to 2048 depending on channel bandwidth from 1.4 MHz to 20 MHz.

Bearing Fault Detection in Condition Monitoring

Predictive Maintenance

Emerson's AMS 9420 wireless vibration transmitter samples a MEMS accelerometer at 25.6 kHz and computes a 6400-line FFT. The frequency spectrum is compared against bearing fault frequency tables derived from the ball pass frequency outer race (BPFO) formula. A peak at the predicted BPFO frequency, typically between 80 Hz and 400 Hz for industrial motors, indicates an outer-race defect before mechanical failure occurs.

How it works in practice

In a practical FFT implementation on a DSP, the input buffer collects N samples at a uniform sampling rate Fs. Before computing the FFT, a window function such as Hanning or Blackman is applied element-wise to the buffer. Windowing reduces spectral leakage that occurs when the signal frequency is not an integer multiple of the frequency resolution bin width, which equals Fs divided by N. The Cooley-Tukey radix-2 FFT divides the N-point transform into two N/2-point transforms, then combines the results using butterfly operations that multiply by complex twiddle factors. For N equals 1024 and Fs equals 44100 Hz, the frequency resolution is 43 Hz per bin and the computation requires approximately 5120 complex multiplications. The magnitude spectrum, computed as the absolute value of each complex bin, is then scaled to units of volts RMS or decibels relative to full scale. Phase information is used in applications such as beamforming where the relative phase between two microphone channels encodes the angle of arrival of a sound source.

Examples

FFT-Based Audio Equaliser on STM32

An STM32H7 reads audio from an I2S microphone at 48 kHz with 32-bit depth. A 1024-point FFT is computed using the CMSIS-DSP arm_cfft_f32 function in about 0.3 ms. The magnitude of each FFT bin is scaled by a gain coefficient stored in a 513-element array representing the equaliser curve. The inverse FFT reconstructs the filtered time-domain signal, which is sent to a PCM5122 DAC for playback. The frequency resolution of 46.9 Hz per bin provides sufficient granularity for 10-band equalisation across 20 Hz to 20 kHz.

OFDM Subcarrier Recovery in LTE Receiver

An LTE base station transmits a 10 MHz signal using 600 active subcarriers out of a 1024-point IFFT. The UE receiver samples the signal at 15.36 MSPS, removes the cyclic prefix of 72 samples, and feeds the remaining 1024 samples into an FFT. The output of the FFT produces 1024 complex values; the 600 values at the active subcarrier indices carry the modulated data symbols. Each symbol is then divided by the channel estimate at that subcarrier to equalise multipath distortion, a single complex division per subcarrier that is possible only because OFDM confines intersymbol interference to the cyclic prefix guard interval.

Future scope

Compressive sensing, based on the observation that many real-world signals are sparse in the Fourier domain, allows reconstruction of a spectrum from far fewer samples than the Nyquist rate requires. This is enabling sub-Nyquist ADC architectures for wideband spectrum monitoring in cognitive radio and electronic warfare systems. Quantum Fourier transforms, which underpin Shor's factoring algorithm, are expected to run on error-corrected quantum processors within this decade and would break RSA encryption, driving migration to post-quantum cryptographic standards. In MRI, non-Cartesian k-space trajectories combined with non-uniform FFT algorithms are reducing scan time for cardiac imaging sequences where motion artefacts limit the usable data acquisition window.