Passband ripple and stopband attenuation resonances occur in certain higher order filters.
Bandpass, bandstop, and hi/low pass filters all exhibit these features in their transfer functions and S-parameters.
Some passband ripple and stopband peaks may not produce the gain or attenuation you need for an RF filter, and a different topology may be required.
RF filters can exhibit passband ripple and stopband attenuation peaks, depending on their topology.
No filter or amplifier is perfect, and systems designers need to carefully select filter topologies to ensure signals are output with the desired phase and amplitude. These circuits are the basic building blocks of analog systems, especially in the RF domain, and they need to be carefully designed in terms of their transfer function, input impedance, output impedance, and S-parameters.
Two features that appear in a filter’s transfer function and S-parameters are passband ripple and stopband attenuation resonances. Analog/RF systems need to have filters and S-parameters that are precisely engineered with the right rolloff and gain, while accommodating passband ripple and stopband resonances. Here’s how these features arise in higher order filters and how they affect signal behavior in your analog system.
Passband Ripple and Stopband Attenuation Peaks
Passband ripple occurs in the high-gain region of a higher-order filter or amplifier’s transfer function, and looks like some variations in the output gain. The same applies to the phase on the output. In effect, the two are not smooth functions of frequency. Ripple can also appear in the stopband in these circuits. The stopband attenuation spectrum can exhibit multiple peaks that resemble passband ripple, and the features are sometimes symmetric.
Just like the transfer function, these features also appear in the S-parameters. Since a transfer function can be calculated from S-parameters, this shouldn’t be surprising. The S-parameters for your circuit can be calculated by looking at a higher-order filter/amplifier as a set of cascaded networks, where the T-parameters for each network are multiplied together. Alternatively, you can use a circuit simulator to calculate the S-parameters for the circuit and plot them in the frequency domain.
To see how passband ripple affects signal behavior, consider the 5th-order elliptic highpass filter shown below. In this higher-order filter, the cutoff for the passband is at 100 MHz, and two peaks can be seen in the insertion loss (blue). The first peak occurs at 102.4 MHz and corresponds to a high-Q resonance, while the second peak at 139.3 MHz corresponds to a low-Q resonance.
Transfer function for a 5th-order elliptic filter with passband ripple and stopband attenuation peaks.
These resonances arise due to the arrangement of multiple LC networks, which are separated by shunt inductors, as shown in the circuit diagram above. Note that this applies to any filter topology or any other circuit with groups of cascaded RLC networks. In effect, the successive reactive sections of the circuit couple to each other, which creates a complex resonance spectrum with multiple peaks and valleys in the S-parameters. If we use the above data to calculate the transfer function, the same peaks and valleys will be present, corresponding to the poles and zeros in this circuit.
The corresponding return loss (red curve) shows valleys that approach negative infinity at these same frequencies (102.4 MHz and 139.3 MHz). This indicates that at exactly these two frequencies we have perfect impedance matching and maximum power transfer into the load resistor RL. Farther from these two frequencies, we see extreme impedance mismatch and strong reflection, as illustrated from the return loss near 0 dB.
How These Features Affect Useful Bandwidth
The example shown above, in a practical sense, only provides low attenuation at ~102.4 MHz and 139.3 MHz. The passband (shown in the blue S21 curve) extends from 100 MHz to infinity, but low attenuation signal transfer only occurs at these frequencies. Farther from these frequencies but in the passband, signals will experience -10 dB attenuation in this circuit. This limits the useful bandwidth to smaller values than the passband unless the filter is designed with very low ripple.
If you want to prevent this problem, you need to design the filter to have minimum ripple, or you need to use a filter with different topology (such as Chebyeshev or Butterworth filters). Note that a different topology may have different roll-off, or undesired resonances, in the transfer function. Thankfully, there are many RF design textbooks that contain design formulas needed to understand how these features in higher-order filter transfer functions are related. You can then use a SPICE simulator to verify passband ripple and stopband attenuation peaks in your circuit’s transfer functions.
Modeling Filter Responses
When your filter has passband ripple, it can be difficult to see exactly how the ripple and attenuation peaks distort an input wideband signal. For a harmonic frequency, the result is rather simple; the output will be scaled by the magnitude of the transfer function and shifted in time by the phase of the transfer function at that frequency, regardless of any passband ripple. For any other signal where power is distributed over a range of frequencies, you need to use a more sophisticated method to see how the circuit affects the output signal.
The simplest method is to calculate the Fourier components of the input signal and multiply each component by the gain and phase factor at that specific frequency. Unfortunately, this simplistic method does not preserve all the information of the input signal, as the input Fourier spectrum needs to be cut off at some higher harmonic (digital designers usually use the 3rd or 5th harmonic). Once you look at multilevel bit streams, there is plenty of ambiguity in the signal bandwidth due to the zeroes in the PSD of the signal.
PAM4 and NRZ power spectra showing differences in useful bandwidth for each signal. Designers need to choose the relevant bandwidth of the signal when determining their desired filter response.
To model a circuit’s response to any stimulus with defined Fourier transform, use the following process:
Calculate the transfer function directly, or from the circuit’s S-parameters.
Take an inverse Fourier transform to get the impulse response function (IRF).
Calculate the time-domain convolution between the IRF and the input signal.
If you’re not a fan of calculating convolutions, you can use the convolution theorem with the Fourier transform of your input signal to predict the output signal from your filter or amplifier circuit. Once you know the impulse response function or the transfer function, you know everything about your circuit and how it will pass or distort input signals. You can then calculate the circuit’s response as long as you can determine the circuit’s transfer function.