Relationship between the s-plane and the z-plane
We can extend the above procedure to the case where Ha(s) is given as a sum of N terms with distinct poles as
For this case the impulse invariant design, H(z), is given by
where L-1 means Laplace inverse. We observe that a pole at s=-αk in the s-plane gives rise to a pole at z = e-αkT in the z-plane and the coefficients in the partial fraction expansion of Ha(s) and H(z) are equal. If the analog filter is stable, corresponding to -kα
being in the left half plane, then the magnitude of e-αkT will be less than unity, so that the corresponding pole of the digital filter is inside the unit circle, and as a result the digital filter also is stable.
While the poles in the s-plane "map" to poles in the z-plane according to the relationship z =esT, it is important to recognize that the impulse invariance design procedure does not correspond to a mapping (transformation) of the s-plane to the z-plane by that relationship or in fact by any relationship. (An example of a transformation is where we actually make a substitution, say,
which, of course, is the bilinear transformation). For example, the zeros of Ha(s) do not map to zeros of H(z) according to this relation. See also matched z-transform later.
We can explore the relationship z=est keeping in mind that it only applies to poles and that it is not a transformation. Set s = σ + jΩ and z = rejω in z = esT to get rejw= e(s + jΩ)T = eσT e jΩT so that r = eσT and ω = ΩT.
The above equations may be used to define that poles in the left half of the final strip in the s-plane map into poles within the single circle in the z-plane as given in the figure for s = s1.
Mapping of poles, z = e s T
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s-plane pole
s = σ + jΩ
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z-plane pole
z = e s T = r e jw
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r
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ω
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0
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1
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1
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0
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jΩs/2
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-1
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1
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π
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-∞ + jΩs/2
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-0
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0
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π
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-∞ - jΩs/2
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-0
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0
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π
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-jΩs/2
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-1
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1
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π
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For s1 = σ1 + jΩ1 we have r = eσ1 T and ω = Ω1T. However, poles at s2 and s3 (which are a distance Ωs from s1) also will be mapped to the same pole that s1 is mapped to. In fact, an infinite number of s-plane poles will be mapped to the same z-plane pole in a many-to-one relationship. These frequencies differ by Ωs = 2πFs = 2π/T (Fs is the sampling frequency in Hertz). This is called aliasing (of the poles) and is a drawback of the impulse-invariant design. The analog system poles will not be aliased in this manner if, in the first place, they are confined to the "primary strip" of width Ωs = 2πFs = 2π/T in the s-plane.
In a similar fashion poles located in the right half of the primary strip in the s-plane will be mapped to the outside of the unit circle in the z-plane. Here again the mapping of the s-plane poles to the z-plane poles is many-to-one
Owing to the aliasing, the impulse invariant design is suitable for the design of low pass and band pass filters but not for high pass filters.
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