Stopband Rejection in a Front-End Filter: How Much Is Enough?
A +23 dBm transmit signal falls to −27 dBm after 50 dB of isolation, and 40 dB of filter rejection takes it to −67 dBm. Required rejection is the leakage level minus the level the receiver can tolerate.
For practitioners
What stopband rejection means
Stopband rejection is the attenuation, in dB, that a filter provides at frequencies outside its passband. The passband is where the filter should pass the signal with little loss. The stopband is where it should block.
Attenuation in dB is 10 log10 of the input-to-output power ratio:
So 40 dB means a power ratio of 10,000. Each 10 dB is a factor of 10 in power. A filter with 40 dB of rejection lets one ten-thousandth of the power through at that frequency.
A filter is not judged by one number. Rejection must be deep at the specific frequencies where a strong unwanted signal sits. At other stopband frequencies a smaller number may be enough.
Why a handset needs it: transmit leakage
A handset transmits and receives through the same front end. The transmitter can output +23 dBm. The receiver input may be looking for signals that are tiny by comparison. Even a small fraction of the transmit signal reaching the receiver input can raise its noise floor or overload it.
The transmit signal does not reach the receiver by one route only. It can pass through the duplexing or switching structure, through board coupling, and through the shared antenna path. Each route has some isolation, which is the attenuation between the transmit port and the receive port. The filter adds rejection at the unwanted frequency on top of that.
Worked example: isolation plus rejection
Take a transmit signal of +23 dBm at the leakage source. Assume 50 dB of isolation between the transmit path and the receive path. The level arriving at the receive filter input is:
Now add a filter with 40 dB of stopband rejection at the transmit frequency, a power ratio of 10,000. The level at the receiver input is:
The total attenuation is 50 dB + 40 dB = 90 dB, which is a power ratio of 10^9. The figure below shows the chain with the level under each node.
The dB values add only under two conditions. Each element must be matched to its source and load impedances, so mismatch interaction between stages is negligible. And no parallel leakage path may bypass the stages. If a second path exists, for example coupling around the filter, the power of the two paths adds at the receiver and the total attenuation is set by the weaker path. A 40 dB filter cannot help if a leakage path with only 30 dB of attenuation goes around it.
How much is enough
The answer is a subtraction. Let P_TX be the transmit level, I the isolation, and P_max the highest unwanted level the receiver can tolerate at that frequency:
The tolerable level P_max comes from the receiver's design and the system requirement. It is not a universal number, and this article does not assume one. If the target is −67 dBm, the required rejection in the example is exactly 40 dB. If isolation drops from 50 dB to 40 dB, the required rejection rises by the same 10 dB, to 50 dB.
Two practical points follow:
- Rejection and isolation trade off one for one in dB. Better isolation lowers the rejection the filter must deliver.
- Cascading helps when the stages are matched. Two 30 dB rejections in series give 60 dB, a power ratio of 1,000,000.
Keep a margin. A filter specified at exactly the minimum leaves nothing for part-to-part spread or for a leakage path you did not model.
Filter order and roll-off
Increasing the order of a filter increases the steepness of the roll-off outside the passband. For an ideal Butterworth lowpass filter of order n and cutoff fc, the attenuation at frequency f is:
For n = 5, at f = 2 fc the term is 2^10 = 1024, so the attenuation is 10 log10(1025) ≈ 30.1 dB. At f = 3 fc it is 3^10 = 59,049, so the attenuation is about 47.7 dB. Far above the cutoff the curve falls by about 20n dB per decade, which is 100 dB per decade for a 5th-order filter.
The cost of a higher order is more elements in the filter. The passband loss usually grows as well, and that loss sits in front of the receiver. Rejection is therefore chosen to meet the requirement, not to be as large as possible.
Reading rejection on a plot
Rejection is read from S21, the forward transmission, in the stopband. A deeper S21 means more rejection. The plot below is computed from the ideal response, not measured data. A real filter will show a floor or ripple set by parasitics and coupling, and a measurement will also be limited by the instrument's own dynamic range. The page on S21 and S11 of a low-pass filter shows how to read these curves.
In plain words · for students and beginners
Stopband rejection in plain words
A door and a loud neighbour
Think of a quiet room and a loud neighbour. The wall between you and the neighbour is the isolation. A thick curtain you hang on your side is the filter. The wall cuts most of the noise. The curtain cuts what is left. Together they cut more than either alone.
This breaks down in one way. If the neighbour's noise also comes in through an open window, the curtain does nothing for that part. A filter cannot block signal that goes around it.
What the curve looks like
A filter lets some frequencies through and blocks others. On a graph, the blocked region is where the line drops far down. The deeper the drop, the more rejection. This curve is computed from an ideal filter, not measured.
The same example in simple steps
- The phone sends a strong signal: +23 dBm. A dBm is a power level; higher means stronger.
- The isolation takes away 50 dB. 23 − 50 = −27 dBm.
- The filter takes away another 40 dB. −27 − 40 = −67 dBm.
- The total taken away is 50 + 40 = 90 dB.
Why can we simply add dB? Because dB count in powers of 10. Adding 10 dB means multiplying the power ratio by 10. So 40 dB is a ratio of 10,000, and adding stages means multiplying their ratios, which is adding their dB. This works only if each stage is matched (it connects cleanly to its neighbours) and no leakage path goes around the stages.
Mistakes beginners make
- Reading the passband loss and thinking it is the rejection. Rejection is measured in the stopband, at the unwanted frequency.
- Asking for the biggest rejection possible. A higher-order filter costs more passband loss, so ask for what the calculation needs.
- Adding dB when a leakage path bypasses a stage. The weaker path then sets the result.
- Mixing up dB and dBm. dB is a ratio between two powers. dBm is a power level.
Remember
- Rejection is attenuation in dB outside the passband; 40 dB means a power ratio of 10,000.
- Required rejection = leakage level after isolation − the level the receiver can tolerate.
- Isolation and rejection add in dB only when stages are matched and nothing bypasses them.
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Frequently asked questions
What is stopband rejection?
It is the attenuation, in dB, that a filter provides at frequencies outside its passband. A larger number means less power gets through at that frequency.
How do I calculate the rejection a filter needs?
Subtract the highest level the receiver can tolerate from the unwanted signal level at the filter input. The difference, in dB, is the minimum rejection at that frequency.
Do isolation and filter rejection add?
Yes, in dB, when each stage is matched to its source and load and no parallel leakage path bypasses them. A 50 dB isolation and a 40 dB rejection give 90 dB in total.
How can I increase the rejection of a filter?
Raise the filter order, which makes the roll-off outside the passband steeper. You can also add a second filter in series, as long as the stages are matched and no leakage path goes around them.