We have obtained a value of system delay by minimizing the integrated squared error between the input signal and the appropriately delayed output signal. Most important, this theoretically obtained value is consistent with previous results and our physical reasoning. The various examples bear this out. The average delay time can also be denormalized for the BP equivalents of the previously discussed signals applied to a BP filter by multiplying by 1/πΔf, where Δf is the BP 3-dB bandwidth in hertz and the LP prototype has unity-radian cutoff frequency. The LP response then corresponds to the envelope of the BP response.

Chapter Contents:

• 8.1 Approximations to a Prescribed Function
• 8.1.1 Method of Moments
• 8.1.2 Approximation of the Rectangular Pulse by the Method of Moments
• 8.1.3 Approximation of the Triangular Function by the Method of Moments
• 8.1.4 Method of Least Squares
• 8.1.5 Least-Squares Approximation for Specified Exponentials
• 8.1.6 Approximation of the Rectangular Pulse by Two Specified Exponentials
• 8.1.7 Least-Squares Approximation for Optimum Amplitudes and Exponents
• 8.1.8 Approximation of the Rectangular Pulse by One Exponential
• 8.1.9 Approximation of the Rectangular Pulse by Two Exponentials
• 8.1.10 Orthogonal Filter
• 8.2 Response of Linear, Time-Invariant Systems to Modulated Waveforms
• 8.2.1 Spectral Approach
• 8.2.2 FM Waveform Analysis by the Spectral Approach
• 8.2.3 Dynamic Approach
• 8.2.4 Quasi-Stationary Analysis
• 8.2.5 Response to an Amplitude-Modulated Signal
• 8.2.6 Response to a Frequency-Modulated Signal
• 8.3 Response of Linear, Time-Variant Systems to Modulated Waveforms
• 8.3.1 Quasi-Stationary Response
• 8.3.2 Quasi-Stationary Analysis of a First-Order System
• 8.3.3 Separable Systems
• 8.3.4 Quasi-Stationary Analysis of Separable Systems
• 8.3.5 Approximation of a Nonseparable System by a Separable System
• 8.3.6 Summary
• 8.4 Average Delay Through a Time-Invariant System
• 8.4.1 Definition of Average Delay
• 8.4.2 Average Delay of a Sinusoid
• 8.4.3 Average Delay of the Impulse Function
• 8.4.4 Average Delay of the Step Function
• 8.4.5 Average Delay of the Rectangular Pulse
• 8.4.6 Summary of Delay Definitions
• References
• Problems

Inspec keywords: band-pass filters; delay filters; time-domain analysis; mean square error methods; low-pass filters

Other keywords: LP response; time-domain operation; system delay; BP equivalent; BP response; BP filter; integrated squared error

Subjects: Interpolation and function approximation (numerical analysis); Filtering methods in signal processing; Signal processing theory; Interpolation and function approximation (numerical analysis)