Digital Signal Sinc Pulse Filtering (Electrical & Computer Engineering)

In digital signal processing (DSP) and high-speed wireless communications (such as 5G base stations and radar transceivers), transmitting sharp rectangular digital pulses through bandwidth-limited channels causes inter-symbol interference. To prevent signal overlap and pulse distortion, engineers shape transmission symbols using an ideal Nyquist filter whose continuous-time impulse response is governed by the normalized sinc function:

where $B = 20.0\text{ MHz}$ is the filter cutoff bandwidth in megahertz and $t$ represents time in microseconds ($\mu\text{s}$).

At the exact center of the transmitted pulse ($t = 0$), direct evaluation of the filter response yields the indeterminate quotient form:

A digital receiver's analog front-end incorporates an automatic gain control (AGC) circuit. To calibrate the receiver's dynamic range without risking ADC (analog-to-digital converter) clipping or numerical division-by-zero exceptions in DSP firmware, the engineering team must determine the precise peak signal voltage and analyze how the signal behaves near the origin.

1. Classify the indeterminate form of $\lim_{t \to 0} h(t)$, verify that the hypotheses of L'Hôpital's Rule are satisfied, and calculate the exact peak normalized impulse response value $h_{\text{peak}} = \lim_{t \to 0} h(t)$.
2. If the transmitted carrier waveform has an amplification factor $A_0 = 3.3\text{ V}$, write the local linear and quadratic Taylor approximations of $V(t) = A_0 \frac{\sin(\pi B t)}{\pi B t}$ around $t = 0$.
3. Explain why embedded digital signal processors cannot directly execute the mathematical expression $\frac{\sin(\pi B t)}{\pi B t}$ at the sampling clock tick $t = 0$, and explain how firmware engineers use the limit evaluated via L'Hôpital's Rule to implement branchless fail-safes in FPGA filter lookup tables.