Digital Signal Sinc Pulse Filtering (Electrical & Computer Engineering) Solution

Problem Overview

We evaluate the peak impulse response $h_{\text{peak}} = \lim_{t \to 0} h(t)$ of an ideal Nyquist sinc filter:

where $B = 20.0\text{ MHz}$ and $t$ is time in $\mu\text{s}$. Direct substitution yields the indeterminate form $\frac{0}{0}$. We resolve the limit using L'Hôpital's Rule, determine peak voltage under amplification $A_0 = 3.3\text{ V}$, and evaluate firmware design strategies.

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Step-by-Step Mathematical Solution

Step 1: Form Classification & L'Hôpital's Rule Application

The governing formulation is L'Hôpital's Rule for Indeterminate Quotients:

(This rule and its hypotheses are defined in L'Hôpital's Rule.)

1. Verify Hypotheses:

   • Let $f(t) = \sin(\pi B t)$ and $g(t) = \pi B t$.
   • Both $f(t)$ and $g(t)$ are smooth, infinitely differentiable functions on $\mathbb{R}$.
   • Evaluating at $t = 0$:
     $$f(0) = \sin(0) = 0, \qquad g(0) = \pi B (0) = 0$$
   • The expression yields the classical indeterminate quotient form $\frac{0}{0}$.
   • The derivative of the denominator $g'(t) = \pi B \neq 0$ for all $t$.
   • Therefore, L'Hôpital's Rule is rigorously valid.

2. Differentiate Numerator and Denominator Independently: Using the Chain Rule on the numerator:

   $$f'(t) = \frac{d}{dt}[\sin(\pi B t)] = \pi B \cos(\pi B t)$$
   Differentiating the linear denominator:
   $$g'(t) = \frac{d}{dt}[\pi B t] = \pi B$$

3. Evaluate the Limit:

   $$\lim_{t \to 0} h(t) = \lim_{t \to 0} \frac{\pi B \cos(\pi B t)}{\pi B} = \lim_{t \to 0} \cos(\pi B t) = \cos(0) = 1$$
   Thus, the normalized peak pulse amplitude is exactly:
   $$h_{\text{peak}} = 1.0$$

Step 2: Scaled Voltage & Local Taylor Approximation

For a transmitted signal scaled by $A_0 = 3.3\text{ V}$:

The peak voltage at the center of the pulse is:

Recall the Taylor series expansion for the sine function: $\sin(u) = u - \frac{u^3}{6} + \mathcal{O}(u^5)$. Substituting $u = \pi B t$:

Therefore, the quadratic approximation of voltage around $t = 0$ is: This confirms that the pulse has a zero first derivative ($V'(0) = 0$, a smooth local maximum) and curves downward quadratically away from the peak.

Step 3: DSP Firmware & Hardware Implementation

1. The Division-by-Zero Trap: In floating-point (IEEE 754) or fixed-point DSP microprocessors (e.g., Texas Instruments TMS320 or ARM Cortex-M), computing $\frac{\sin(0)}{0}$ at runtime produces a NaN (Not a Number) floating-point exception or triggers an integer hardware divide-by-zero interrupt trap, causing firmware crash or signal loss.
2. Branchless Firmware Implementation: Because L'Hôpital's Rule establishes that $h(0) = 1$ is a removable singularity, firmware engineers implement a conditional check or piece-wise ternary operator in digital filter synthesis:

   float sinc_filter(float t, float B) {
       float arg = M_PI * B * t;
       if (fabsf(arg) < 1.0e-7f) {
           return 1.0f - (arg * arg) / 6.0f; // High-precision Taylor expansion avoids division
       }
       return sinf(arg) / arg;
   }
   

   In FPGA hardware (VHDL/Verilog), the central filter tap at index $k = 0$ is hardcoded directly to the exact integer fixed-point representation of $1.0$ (e.g., 0x7FFF in 16-bit Q15 format).

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Engineering Conclusion

In telecommunications and digital signal processing, mathematical removable singularities cannot be ignored at runtime. If an ADC or DSP algorithm attempts naive numerical evaluation of the sinc function at $t = 0$, hardware interrupts or NaN errors crash communications pipelines. L'Hôpital's Rule proves mathematically that the limiting response exists and equals unity ($1.0$).

By recognizing that $\lim_{t \to 0}\frac{\sin(\pi B t)}{\pi B t} = 1$, engineers can safely normalize finite impulse response (FIR) filter tap coefficients, configure ADC voltage references to precisely $3.3\text{ V}$, and embed Taylor-series branchless fallbacks in FPGA firmware to guarantee continuous, zero-dropout wireless transmission.