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The impulse can be modeled as a Dirac delta function for continuous-time systems, or as the discrete unit sample function for discrete-time systems. The Dirac delta represents the limiting case of a pulse made very short in time while maintaining its area or integral (thus giving an infinitely high peak).
If a system initially rests at its equilibrium position, from where it is acted upon by a unit-impulse at the instance t=0, i.e., p(t) in the equation above is a Dirac delta function δ(t), () = | = =, then by solving the differential equation one can get a fundamental solution (known as a unit-impulse response function)
The impulse response can be computed to any desired degree of accuracy by choosing a suitable approximation for δ, and once it is known, it characterizes the system completely. See LTI system theory § Impulse response and convolution. The inverse Fourier transform of the tempered distribution f(ξ) = 1 is the delta function.
Linear interpolation is equivalent to a triangular impulse response; windowed sinc approximates a brick-wall filter (it approaches the desirable brick-wall filter as the number of points increases). The length of the impulse response of the filter in method 1 corresponds to the number of points used in interpolation in method 2.
The FIR convolution is a cross-correlation between the input signal and a time-reversed copy of the impulse response. Therefore, the matched filter's impulse response is "designed" by sampling the known pulse-shape and using those samples in reverse order as the coefficients of the filter. [1]
A sample is a value of the signal at a point in time and/or space; this definition differs from the term's usage in statistics, which refers to a set of such values. [ A ] A sampler is a subsystem or operation that extracts samples from a continuous signal .
where the h[•] sequence is the impulse response, and K is its length. x [•] represents the input sequence being downsampled. In a general purpose processor, after computing y [ n ], the easiest way to compute y [ n +1] is to advance the starting index in the x [•] array by M , and recompute the dot product.
The delayed output makes this a causal system.The impulse response of the delayed FOH does not respond before the input impulse. This kind of delayed piecewise linear reconstruction is physically realizable by implementing a digital filter of gain H(z) = 1 − z −1, applying the output of that digital filter (which is simply x[n]−x[n−1]) to an ideal conventional digital-to-analog ...