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Poles and zeroes

In filter theory we often speak in terms of poles and zeroes. Both words relate to the mathematical functions that describe the properties of filters. If you draw the circuit  of a simple lowpass filter with, say two elements – a series-L and a shunt-C – and derive its transfer function then you will get a polynomial in the complex frequency variable s.

Polynomials are functions in one variable in the form

f(x) = a_0 + a_1 * x + a_2 * x^2 + a_3 * x^3 + .... + a_n * x^n

Clearly, the value of the polynomial for any given value of x depends entirely on the coefficients a_0, a_1, a_2 … a_n. Based on the so-calledfundamental theorem of algebra (J.F. Gauss, 1777–1855) this function can also be expressed in the form of so-called linear factors containing its zeroes

f(x) = a_n * (x  -  x0_1) * (x  -  x0_2) * (x  -  x0_3) * ..... * (x  -  x0_n)

So, if x = x0_1 or x = x0_2 and so on, then the function value becomes zero. Again, the value of the polynomial in its product form depends on the zeroes of the polynomial for any given value of x.

Consider a capacitance parallel to a series-inductance in the LC lowpass filter. At resonance, the reactance is infinite, suppressing all transmission through the lowpass, or: introducing a so-called ‘finite frequency transmission zero’ or, simpler: ‘a notch’. The resulting lowpass transfer function becomes a rational function with a numerator polynomial and a denominator polynomial. The latter being much simpler than the numerator polynomial because it is just creating the ‘notch’ – it becomes zero at the notch frequency.

f(x) = (a_n * (x  -  x0_1) * (x  -  x0_2) * (x  -  x0_3) * ..... * (x  -  x0_n))/(x-y0_1)

The zeroes of the denominator polynomial of the rational function create the poles of the transfer function because 1 / 0 = infinity.

Polynomial coefficients

Given that polynomials can be ‘created’ from their zeroes, it is noteworthy that all polynomial coefficients will be real numbers if the zeroes are  ‘conjugate complex’ pairs:

x0_1 = -1 +j*3            x0_2 = -1 - j*3

Luckily, many of our filter polynomials have zeroes which are conjugate complex and therefore we often deal with polynomials with real coefficients. In filter synthesis we often have convert poles and zeroes to polynomials and vice versa. This is accomplished by multiplying out all the linear factors shown above and the reverse is done by using so-called ‘root finding’ algorithms.

If you really like to dig deep into filter theory, then understanding polynomials and their properties is a good pre-requisite.

 

Transmission zeroes

Filter jargon often uses the word ‘transmission zeroes’. Zeroes of transmission occur in a filter if the input impedance is either zero or infinity. If these two extreme impedances exist, all filter input signals are fully reflected and therefore no transmission can occur: the transmission through the filter is zero. Take for example the lowpass filter with a series-L and shunt-C. Both elements will cause a transmission zero: the series-L has an infinite impedance at f = infinity and the shunt-C has zero impedance at f = infinity. Both elements create extreme impedances at f = infinity. Therefore the lowpass filter must have two transmission zeroes at f = infinity. The idea of an infinitely high frequency seems to be of little practical value, it does however help in understanding a filter’s frequency behavior. The existence of transmission zeroes at f = infinity manifests itself in the filter response and is therefore felt at finite frequencies (f < infinity). Filters whose transfer functions are polynomials (no denominator, example: Chebyshev and Butterworth filters) have all their transmission zeroes at f = infinity.

 

Filter analysis based on poles and zeroes

When the poles and zeroes of the filter polynomials are known, the 2-port parameters of a filter can be calculated directly from the poles and zeroes – including the group delay. This is mathematically relatively simple and straightforward because each pole and each zero makes a contribution to the overall filter function. Therefore, a pole-zero based analysis involves simple products and sums rather than lengthy and complex calculations. The effect of finite Q can also be considered accurately because of the fact that losses simply move the poles and zeroes of the transfer function to the left in the complex plane by a small amount, proportional to the quality factor Q. The frequency dependent insertion loss of narrow bandpass filters in the presence of losses (average resonator Qu) can be calculated with highest accuracy on the basis of the poles and zeroes of the filter transfer function. See example.