Before we consider frequency transform techniques, lets consider the second orderseries-tuned LCR c
We will find that the mathematics for each filter design will be very similar. For example, the difference between a lowpassand highpass filter is essentially an inverse—the frequencies below ωc are mapped into frequencies above ωc—and vice versa
In other words, the transmission through a low-pass filter at one half the cutoff frequency will be equal to the transmission through a (mathematically similar) high-pass filter at twice the cutoff frequency.
Converting a normalized lowpass filter to another lowpass filter allows to set the cutoff frequency of the resulting filter. This is also called frequency scaling.
As an example, the biquadratic transfer functionwill be transformed into:The name biquadratic stems from the fact that the functions has two second order polynoms:
If the filter is given by a circuit and its R, L and C element values found in a table, the transfer function is scaled by changing the element values.The resistance values will stay as they are (a further impedance scaling can be done).The capacitance values are changed according to:The inductance values are changed according to:In the circuit, all capacitances and inductances values are divided by fc
Converting a lowpass filter to a highpass filter is one of the easiest transformations available. To transform to a highpass, we will replace all S in our equation with the following:This operation can be performed using thisMATLAB command:lp2hp
The specification for a high-pass filter includes the passband edge frequency, ωhp,and the stopband edge frequency, ωhs The maximum passband attenuation is Amaxand the minimum stopband attenuation is Amin. The transformation of thehigh-pass specification to an equivalent normalised low-pass specification isachieved by applying the frequency transform SL = 1/s, where SL is the low-passnormalised complex frequency variable. However to account for the process ofnormalization we must replace SL by 1/(s/ω p) =ωp/s.
We calculate the value for ε and n for the filter type chosen using the equivalentnormalised low-pass obtained previously. From these two quantities, we can thenobtain the normalised attenuation function. This must be frequency de-normalisedand the, low-pass to high-pass frequency transformation, performed. To denomalise a Butterworth approximation loss function we used
Frequency Transformationi.e. Low Pass to High Pass Filter etc.
Frequency Transformations• We need to apply a suitable frequency transformation, if we wish to designbandpass, bandstop and high-pass filters, using the low-pass approximatingfunction analysis
Filter Transformations• We can use the concept of filter transformations to determine the new filter designs from a lowpass design. As a result, we can construct a 3rd-order Butterworth high-pass filter or a 5th-order Chebychev bandpass filter!
Normalized Lowpass Filter• When designing a filter, it is common practice to first design a normalized low- pass filter, and then use a spectral transform to transform that low-pass filter into a different type of filter (high-pass, band-pass, band-stop).• The reason for this is because the necessary values for designing lowpass filters are extensively described and tabulated. From this, filter design can be reduced to the task of looking up the appropriate values in a table, and then transforming the filter to meet the specific needs
Lowpass to Lowpass Transformation• Having a normalized transfer function, with cutoff frequency of 1 Hz, one can modify it in order to move the cutoff frequency to a specified value
Lowpass to Highpass we define values for the transformed frequency Ω as Equivalent LPF response
Lowpass to Highpassbut with frequency transformation, we substitute for ω/ωp with In the Chebychev case, we apply the substitution
Conversion of Low-pass and High-passFilter transfer functions fromcontinuous time to discrete timedifference equations.• The following converts two filter transfer function that are represented in the Laplace Space• (Continuous time) into their discrete time equivalents in the Z-space using the Bilinear Transform• (AKA Tustin’s Method), then converts them to difference equations expressing the current output as a combination of previous inputs and outputs.