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Example https://www.bsse.ethz.ch/bel/research/cmos-microsystems/microelectrode-systems/ _jcr_content/par/fullwidthimage/image.imageformat.fullwidth.340125254.png Prof. Andreas Hierlemann's Group Simulating noise (Discrete samples) Background on Fourier Transforms Reference: https://web.eecs.umich.edu/~fessler/course/451/l/pdf/c5.pdf Accessed on 07 August 2018 Fourier Transform Discrete Time Fourier Transform Discrete Fourier Transform Computing PSD Spectral leakage and Windowing Modules Page 2

Spectral leakage and Windowing Multiplication Convolution in Frequency in time To ensure no spectral leakage Example 1: Fs = 100Hz, Fin = 2Hz, N = 100 Modules Page 3

Repeating the above sequence of N (100) -samples Example 2: Fs = 100Hz, Fin = 2.1Hz, N = 100 Modules Page 4

Repeating the above sequence of N (100) -samples Example 2: Fs = 100Hz, Fin = 2.1Hz, N = 100 Windowing What is the problem? Spectral leakage 1. The signal may not lie is the desired frequency bin ( Fin/Fs ≠ m/N) a. Asynchronous sampling b. Signal frequency is unknown 2. Additive noise a. Periodic repetition by DFT introduces kinks --> leakage What is the solution? Basic idea: Less weight to signal in the ends: See signal through a window. Examples of windows R T H Modules Page 4

H B See Window_Example.m and PSDCalculator2. Modules Page 5

Module-2e: Front-End for Detecting Pulses 26 July 2018 11:58 Examples of pulsed signals in sensor ICs Lopez, Carolina Mora, et al. "An implantable 455-active-electrode 52-channel CMOS neural probe." IEEE Journal of Solid-State Circuits 49.1 (2014): 248-261. Braga, Leo Huf Campos, et al. "A Fully Digital 8× 16 SiPM Array for PET Applications With Per-Pixel TDCs and Real-Time Energy Output." J. Solid-State Circuits 49.1 (2014): 301-314. Modules Page 1

Koutsos, Ermis, Vlad Cretu, and Pantelis Georgiou. "A Muscle Fibre Conduction Velocity Tracking ASIC for Local Fatigue Monitoring." IEEE Trans. Biomed. Circuits and Systems 10.6 (2016): 1119-1128. When signals are random pulses in time: How to define SNR? • What is optimum SNR? • What is the right bandwidth? • Since we know pulse shape, can we improve SNR further? • Case 1: Sufficient SNR Higher SNR case Modules Page 2

Higher SNR case Receiver Operating Characteristic See Matlab example: Amplitude_Thresholding.m for noise (sigma = 0.1 and 0.2) Modules Page 3