9/27/2007 Filters notes 1/2 Jim Stiles The Univ. of Kansas Dept. of EECS H. Filters Recall that the electromagnetic spectrum is generally full of signals at all different frequencies. Somehow, we have to reject all those signals—except for the one signal we are interested in! Q: Yikes! How can we possibly do that? A: Each signal has its own “IP address”—its carrier frequency 0ω ! We can build devices that select or reject signals based on this carrier frequency. We call these devices filters. HO: Filters HO: The Filter Bandwidth HO: The Filter Phase Function Q: Why do we give a darn about phase function ()21S∠ω? After all, phase doesn’t matter. A: Phase doesn’t matter!?! A typical rookie mistake! HO: Filter Dispersion HO: The Linear Phase Filter9/27/2007 Filters notes 2/2 Jim Stiles The Univ. of Kansas Dept. of EECS Q: So how do we specify a microwave filter? How close to an ideal filter can we build? A: HO: Microwave Filter Design Q: How do I decide what filter type and order to use? A: HO: The Filter Design Worksheet Let’s summarize what we’ve learned! HO: The Microwave Filter Spec Sheet9/27/2007 Filters 1/8 Jim Stiles The Univ. of Kansas Dept. of EECS Filters A RF/microwave filter is (typically) a passive, reciprocal, 2-port linear device. If port 2 of this device is terminated in a matched load, then we can relate the incident and output power as: 221out incPSP= We define this power transmission through a filter in terms of the power transmission coefficient T: 221outincPSP=T  Since microwave filters are typically passive, we find that: 01≤≤Τ in other words, out incPP≤ . Filter incPoutP9/27/2007 Filters 2/8 Jim Stiles The Univ. of Kansas Dept. of EECS Q: What happens to the “missing” power inc outPP−? A: Two possibilities: the power is either absorbed (Pabs) by the filter (converted to heat), or is reflected (Pr) at the input port. I.E.: Thus, by conservation of energy: inc r outabsPPPP=++ Now ideally, a microwave filter is lossless, therefore 0absP= and: inc r outPPP=+ which alternatively can be written as: 1inc r outinc incroutinc incPPPPPPPPP+==+ Filter incPoutP rPabsP9/27/2007 Filters 3/8 Jim Stiles The Univ. of Kansas Dept. of EECS Recall that out incPP= Τ , and we can likewise define rincPP as the power reflection coefficient: 211rincPSP=Γ  We again emphasize that the filter output port is terminated in a matched load. Thus, we can conclude that for a lossless filter: 1=+ΓΤ Which is simply another way of saying for a lossless device that 2211 211SS=+. Now, here’s the important part! For a microwave filter, the coefficients Γ and Τ are functions of frequency! I.E.,: ()ωΓ and ()ωΤ The behavior of a microwave filter is described by these functions!9/27/2007 Filters 4/8 Jim Stiles The Univ. of Kansas Dept. of EECS We find that for most signal frequencies sω, these functions will have a value equal to one of two different approximate values. Either: ()0sωω=≈Γ and ()1sωω=≈Τ or ()1sωω=≈Γ and ()0sωω=≈Τ In the first case, the signal frequency sω is said to lie in the pass-band of the filter. Almost all of the incident signal power will pass through the filter. In the second case, the signal frequency sω is said to lie in the stop-band of the filter. Almost all of the incident signal power will be reflected at the input—almost no power will appear at the filter output.9/27/2007 Filters 5/8 Jim Stiles The Univ. of Kansas Dept. of EECS Consider then these four types of functions of ()ωΓ and ()ωΤ : 1. Low-Pass Filter Note for this filter: () ()1001ccccωωωωωωωωωω≈< ≈ <⎧⎧⎪⎪==⎨⎨⎪⎪≈> ≈>⎩⎩ΤΓ This filter is a low-pass type, as it “passes” signals with frequencies less than cω, while “rejecting” signals at frequencies greater than cω. ()ωΓ ()ωΤ ω ω cω cω 1 1 Q: This frequency cω seems to be very important! What is it?9/27/2007 Filters 6/8 Jim Stiles The Univ. of Kansas Dept. of EECS A: Frequency cω is a filter parameter known as the cutoff frequency; a value that approximately defines the frequency region where the filter pass-band transitions into the filter stop band. According, this frequency is defined as the frequency where the power transmission coefficient is equal to ½: ()0.5cωω==Τ Note for a lossless filter, the cutoff frequency is likewise the value where the power reflection coefficient is ½: ()0.5cωω==Γ 2. High-Pass Filter ()ωΓ ()ωΤ ω ω cω cω 1 19/27/2007 Filters 7/8 Jim Stiles The Univ. of Kansas Dept. of EECS Note for this filter: () ()0110ccccωωωωωωωωωω≈< ≈<⎧⎧⎪⎪==⎨⎨⎪⎪≈> ≈ >⎩⎩ΤΓ This filter is a high-pass type, as it “passes” signals with frequencies greater than cω, while “rejecting” signals at frequencies less than cω. 3. Band-Pass Filter Note for this filter: () ()00001 ∆ 20∆ 20 ∆ 21∆ 2ωω ω ωω ωωωωω ω ωω ω≈−< ≈ −<⎧⎧⎪⎪==⎨⎨⎪⎪≈−> ≈−<⎩⎩ΤΓ ()ωΓ ()ωΤ ω ω 0ω 0ω ∆ω ∆ω 1 19/27/2007 Filters 8/8 Jim Stiles The Univ. of Kansas Dept. of EECS This filter is a band-pass type, as it “passes” signals within a frequency bandwidth ∆ω, while “rejecting” signals at all frequencies outside this bandwidth. In addition to filter bandwidth ∆ω, a fundamental parameter of bandpass filters is 0ω, which defines the center frequency of the filter bandwidth. 3. Band-Stop Filter Note for this filter: () ()00000 ∆ 21∆ 21 ∆ 20∆ 2ωω ω ωω ωωωωω ω ωω ω≈−< ≈−<⎧⎧⎪⎪==⎨⎨⎪⎪≈−> ≈ −<⎩⎩ΤΓ This filter is a band-stop type, as it “rejects” signals within a frequency bandwidth ∆ω, while “passing” signals at all frequencies outside this bandwidth. ()ωΤ ()ωΓ ω ω 0ω 0ω ∆ω ∆ω 1 19/27/2007 Filter Bandwidth 1/3 Jim Stiles The Univ. of Kansas Dept. of EECS ()fΤ 0f Lf Hf f∆ Filter Bandwidth We find that the vast majority of filters in microwave receivers are of the bandpass variety. Some of these bandpass filters will likely have a fairly wide-bandwidth, a bandwidth specified by the lowest frequency that resides within the passband ()Lf and the highest frequency that resides within the passband ()Hf. Other bandpass