Spike-based encodings are sparse and energy-efficient. However, previous theory for such spiking representations is largely disconnected from most signal processing literature and does not extend to multi-scale frames. We recast spike-encoders as time-causal overcomplete wavelet frames with closed-form bandwidth tiling and reconstruction error bounds. We show that scale covariance, frame bounds, and reconstruction guarantees hold for any finite-energy input in both encoding and decoding steps. The proposed spiking bandpass wavelets preserve the sparsity and locality of spiking representations: A threshold parameter sets the event rate, and the reconstruction error grows linearly with it, up to spike quantization and time discretization. On ECG and audio datasets, we reach reconstruction accuracy comparable to continuous wavelet transforms, multi-channel integrate-and-fire time encoding machines, and Sigma-Delta modulation. The experiments demonstrate robustness of the proposed spiking bandpass wavelets to noise and spike-time jitter, and that the spike activity shifts across channels as predicted by the covariance property under temporal rescaling. Every operation reduces to a leaky integrate-and-fire difference with thresholding, and both the encoder and the decoder map directly to existing neuromorphic hardware.

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