Short-time Fourier Transform-based Signal Recovery for Modulo Analog-to-Digital Converters

TL;DR

This work presents a short-time Fourier transform (STFT) based recovery method for signals encoded with modulo analog-to-digital converters that provide 1-bit folding information. By performing unfolding on short, overlapping frames and using a tapered window to mitigate spectral leakage, the approach achieves low latency while maintaining reconstruction accuracy, under explicit oversampling and quantization constraints with a formal mean-squared-error guarantee. Compared with higher-order difference and full-window Fourier methods, the STFT-based method reduces computational complexity and improves robustness in low-resolution, low-sampling regimes, enabling modulo-ADC advantages over conventional ADCs in oversampled settings. Numerical results corroborate the theoretical MSE bounds, illustrate the impact of spectral leakage, and show the method outperforms HoD-based recovery at practical operating points.

Abstract

This study introduces a short-time Fourier transform-based method for reconstructing signals encoded using modulo analog-to-digital converters with 1-bit folding information. In contrast to existing Fourier-based reconstruction approaches that require complete access to the entire observation, the proposed technique performs reconstruction over short, overlapping segments, enabling significantly lower latency while preserving the recovery accuracy. We also address the spectral leakage introduced by the windowing operation by selecting window parameters that balance the leakage suppression and the computational complexity of the algorithm. In addition, we establish conditions under which the correct unfolding of the modulo samples is guaranteed, leading to a reconstruction error determined solely by the quantization noise at the output. The numerical results demonstrate that the proposed method enables modulo analog-to-digital converters to surpass the mean squared error performance of conventional analog-to-digital converters. Furthermore, the proposed recovery method offers improved reconstruction performance compared with higher-order difference-based recovery, particularly in low-resolution and low-sampling rate regimes.

Figures (11)

• Figure 1: Schematic diagram of the modulo ADC with 1-bit folding information. The input signal is wrapped by the modulo operation whenever it exceeds the converter’s dynamic range, producing a folded output sequence. A folding detector simultaneously generates a continuous-time folding signal $c(t)$, and the corresponding discrete-time folding information $c[n]$ is obtained by sampling this signal.
• Figure 2: Modulo operator mechanism to generate the folded signal $f_{\lambda'}(t)$ and the folding information signal $c(t)$. The comparators detect folding events and provide control signals to the discrete voltage generator that produces the wrapped output. At the same time, comparator outputs are processed to generate the continuous-time folding information $c(t)$.
• Figure 3: Overview of the proposed STFT-based recovery method for modulo sampling. The proposed recovery method is composed of three steps: (1) out-of-band DFT computation, (2) estimation of the modulo residue via 1-bit folding information and out-of-band DFT values, and (3) removal of modulo residue from folded signal.
• Figure 4: Numerical example illustrating the construction of $\mathbf{V}_{\mathcal{S}_{i}}$ for $N = 64$, $\mathrm{OF} = 4$, and $\alpha = 0.5$. The top-left plot shows the input signal $f(t)$ and the corresponding modulo ADC output $f_{\lambda'}[n]$ when $\lambda' = 0.30$ and no quantization. The folding events are indicated by $c[n]$ in the bottom–left plot. The top–right plot presents the DTFT magnitude spectrum of $f[n] = f(nT_{\mathrm{s}})$ and $w[n]\cdot\underline{f}[n] = w[n]\cdot (f[n] - f[n-1])$. The spectral leakage is visible due to the finite window length. Components below -60 dB are assumed to have negligible impact on the computation. The length-$\delta_{\mathrm{SL}}$ intervals marked in the top-right plot denote the OOB frequency regions whose leakage exceeds -60 dB. These regions correspond to the $K_{\mathrm{SL}}$ discrete frequency indices marked in the DFT spectrum of the (windowed) 1st-order difference of folded signal $w[n]\cdot \underline{f_{\lambda'}}[n]$, as shown in the bottom–right plot. The matrix $\mathbf{V}_{\mathcal{S}_{i}}$ is constructed by selecting the rows and columns of a DFT matrix corresponding to the OOB discrete frequency indices and folding indices, respectively.
• Figure 5: Illustration of the scaling correction for $\underline{\tilde{z}_{w}^{(i)}}[n]$. The top plot shows the original signal $f(t)$ and the folded samples of the modulo ADC for the $(i-1)$-th and $i$-th segments, denoted as $f_{\lambda',\mathrm{q}}^{(i-1)}[n]$ (blue) and $f_{\lambda',\mathrm{q}}^{(i)}[n]$ (red), respectively. The dashed constant lines correspond to the modulo thresholds. The bottom plot shows the first-order differences of the (windowed) modulo pre-estimates in the $(i-1)$-th and $i$-th segments, denoted $\underline{\tilde{z}_{w}^{(i-1)}}[n]$ (blue) and $\underline{\tilde{z}_{w}^{(i)}}[n]$ (red), respectively. Due to the tapered cosine window, the recovered amplitudes of $\underline{\tilde{z}_{w}^{(i-1)}}[n]$ and $\underline{\tilde{z}_{w}^{(i)}}[n]$ in the overlap region (dash-dot box) are attenuated. However, adding the last $\alpha N/2$ samples of $\underline{\tilde{z}_{w}^{(i-1)}}[n]$ to the first $\alpha N/2$ samples of $\underline{\tilde{z}_{w}^{(i)}}[n]$ corrects the amplitude of $\underline{\tilde{z}_{w}^{(i)}}[n]$.

Theorems & Definitions (6)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1
• proof