Wideband Spectrogram

The fundamental trade-off between time resolution and frequency resolution governed by the uncertainty principle. A shorter FFT window provides better time localization to pinpoint transient events but results in coarser frequency bins. A longer window provides fine spectral detail but smears short-duration pulses over time.

• Narrowband spectrogram: Long window, fine frequency resolution, poor time resolution.
• Wideband spectrogram: Short window, fine time resolution, poor frequency resolution.
• The product of time resolution and bandwidth is a constant.

The choice of window function (Hamming, Hann, Blackman, Kaiser) applied before the FFT directly controls spectral leakage. A rectangular window offers the best frequency resolution but introduces high-level sidelobes that can mask weak signals near strong ones.

• Hann/Hamming: Good sidelobe suppression, moderate main lobe width.
• Blackman: Excellent sidelobe suppression, wider main lobe.
• Kaiser: Adjustable parameter (beta) to trade off sidelobe level for main lobe width.
• Windowing is mandatory for analyzing non-coherent signals.

To capture events that fall on FFT boundaries, consecutive transforms are overlapped, typically by 50% or 75%. Overlap processing ensures no signal energy is lost due to window tapering at the edges.

• 50% overlap: Standard for Hann windows, provides uniform weighting.
• 75% overlap: Provides smoother time evolution, at the cost of higher computational load.
• Without overlap, short-duration pulses occurring at the boundary between two FFTs can be severely attenuated.

The displayed dynamic range is the difference between the noise floor and the maximum signal level represented in the visualization. A logarithmic power scale (dBm or dBFS) is essential to compress large amplitude variations.

• Colormap selection (e.g., 'hot', 'jet', 'inferno') maps power levels to colors, highlighting weak signals against the noise.
• A typical display dynamic range is 60-80 dB.
• Persistence or averaging modes can reveal hidden signals below the instantaneous noise floor.

Resolution Bandwidth is the width of a single frequency bin, determined by the sample rate and the FFT size (RBW = Fs / N_FFT). It defines the minimum frequency separation required to distinguish two adjacent carriers.

• A smaller RBW (larger FFT) resolves closely spaced tones but slows the update rate.
• A wider RBW (smaller FFT) provides a faster update rate for detecting agile frequency hoppers.
• RBW directly impacts the visibility of spectral lines versus broadband noise.

Time resolution is the minimum duration of a transient event that can be distinctly observed. It is equal to the effective FFT window length. The update rate is how often a new spectral line is computed and displayed.

• For a 1024-point FFT at 100 MHz sample rate, the time resolution is 10.24 µs.
• Overlap processing increases the update rate without changing the time resolution.
• High update rates are critical for visualizing frequency-hopping spread spectrum signals.

The smearing of energy from a strong signal into adjacent FFT bins, obscuring weaker signals. Caused by analyzing a non-integer number of signal cycles. Mitigated by applying a window function (e.g., Hamming, Blackman, Kaiser) before the FFT to taper the signal at the boundaries.

• Rectangular window: Best frequency resolution, worst leakage (-13 dB sidelobes)
• Blackman-Harris window: Excellent sidelobe suppression (-92 dB) for high dynamic range
• Trade-off: Better leakage performance always reduces frequency resolution (widens the main lobe)

To capture transient events shorter than a single FFT frame, consecutive FFTs are overlapped in time. Without overlap, a pulse occurring on a frame boundary can be split and attenuated.

• 50% overlap: Standard for most spectral monitoring; doubles the effective time resolution
• 75% overlap: Used for high-resolution time-frequency analysis of agile emitters
• Overlap-add/Overlap-save: Methods for block-based convolution that directly relate to how streaming spectrograms are constructed in real-time pipelines

The essential preprocessing step before spectrogram generation. A DDC translates a digitized wideband signal from a high sample rate to a lower, complex baseband representation through mixing, filtering, and decimation.

• Numerically Controlled Oscillator (NCO): Tunes to the center frequency of interest
• Decimation factor: Determines the output sample rate and thus the spectrogram's Nyquist bandwidth
• Modern wideband receivers use cascaded DDC chains to simultaneously generate multiple narrowband spectrograms from a single wideband input

Instead of a single wideband FFT, a polyphase filter bank splits the spectrum into multiple uniformly spaced sub-bands, each processed independently. This provides superior channel isolation compared to a raw FFT.

• Prototype filter: A low-pass FIR filter designed for high stopband attenuation
• Polyphase decomposition: Rearranges the prototype filter to share computation across all channels
• The result is a spectrogram where each frequency bin has a flat passband and steep roll-off, eliminating the scalloping loss inherent in a plain FFT

A waterfall plot is the standard visualization of a spectrogram, scrolling FFT frames vertically over time with color representing power. Persistence overlays multiple frames using decay or histogram statistics to reveal infrequent events.

• Infinite persistence: Retains the maximum power ever observed per bin, useful for capturing rare emitters
• Probability density: Displays the statistical distribution of power levels, distinguishing constant carriers from bursty traffic
• Implemented in hardware using ping-pong buffers to stream new FFT data into the display without tearing

Generating high-resolution spectrograms on FPGAs requires mapping floating-point algorithms to fixed-point integer arithmetic. This introduces quantization noise that can mask weak signals if not managed.

• Q-format notation: Q15 for 16-bit signed fractional, Q0.15 for pure fractional
• Bit growth management: Each FFT stage (butterfly) increases bit width; truncation or rounding must be applied strategically
• Block Floating-Point (BFP): A compromise where a single exponent is shared across an entire FFT frame, preserving dynamic range without full floating-point hardware cost