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Recently, I stumbled upon a stepped-triangular (ST) approximation that can be implemented as a cascade of recursive running sum (RRS) filters. The following is a short introduction to the stepped-triangular approximation.



The stepped-triangular approximation was introduced by Jovanovic-Dolecek and Mitra [1] as a quantized approximation of a low-pass filter (LPF). Figure 1 shows an example of the approximation.

[Figure 1: Stepped Approximation of a LPF Impulse]

The original paper [1] introduced a method for the stepped-triangular approximation for a LPF impulse response. And the resulting transfer function is as a cascade of recursive sums,

The H rrs (z) is the familiar recursive running sum,

which has a flat impulse response,

[Figure 2: Impulse Response Recursive Running Sum]

H rss (z) is a recursive sparse sum where every s samples are used,

[Figure 3: Impulse Response Recursive Sparse Sum]

Both of the following can be implemented with an efficient recursive form that contains an integrator and difference combination.



To generate the stepped-triangular approximation, from a LPF impulse response, a simple rounding function g[n] is used,

The rounding function g[n] will quantize h[n] into one of three stepped-triangular approximation types. Figures 1 and 4 show one such an example. The three types determine the number of samples at the peak, less than s, equal to s, or greater than s, where s is the number of samples per step.

[Figure 4: Frequency Response for an ST-Approximation]

Figure 4 is an example of a ST-approximation frequency response. I will leave it up to the reader to decide if this is a good approximation given the gains in computation efficiency. The blue is the original frequency response and the red is the approximation using the stepped-triangual approach.

[1] Jovanovic-Dolecek, Gordana, Mitra, Sanjit, “Design of FIR Lowpass filters using stepped triangular approximation”