# Designing a filter to achieve a specific frequency response

I am attempting to design a circuit which filters and amplifies an analog signal. The goal is to achieve the frequency response in the figure below.

I planned to use a second order active low pass filter with a gain of 24(A1) and a second order active high pass filter with a gain of 2(A2). However, my professor has recently given me feedback regarding my proposed filter design which simply stated "This will not work".

I understand this graph appeared in another question but I could not understand what was being asked so I will simply list my questions.

1) Why would my proposed filter design not achieve the required frequency response?

2) What other circuits could produce this frequency response?

Note: I have looked into summing amplifiers but I am not sure how you could separate the HPF and LPF to match the gains at A1 and A2 without the signal bypassing the filter that blocks it since the summing amplifiers input branches would be in parallel.

The only restriction is we can't use digital electronics.

• are you sure you need to achieve exactly that shape (which is damn near impossible in analog filtering), or is it more that you should have gain $A_1$ in passband, and gain $\le A_2$ in stopband? In that case, a simple low pass filter would do. It'd be a different interpretation of the figure than you do – but it seems more practical for a design excercise, especially when you're just given one continuous line; usually, when specifying a filter, an engineer would mark the pass- and stopbands with their respective gains / attenuations, and make sure not to specify anything about the… – Marcus Müller Apr 17 '17 at 12:54
• … spaces between, so that a filter designer would have the freedom to put transition bands there, making the designer's job possible to fulfill. If, on the other hand, this is a spectral mask, i.e. the maximum amount of gain you are allowed at a frequency, the diagram starts making sense and you end up with a simple low pass. – Marcus Müller Apr 17 '17 at 12:57

• 1st order filter is 6dB/octave or 20dB/decade
• 2nd order is 2x above etc.for 3x
• breakpoint is defined by -3dB or 0.707 ratio (rt2)
• 1/2=50% ratio is when X(f)=R in a 1st order filter.

## Design Specs

• A1 (< f1) = 20
• f1 = 400 Hz
• A2 (> f2) = 5
• f2 = 5kHz

• Slope (dB/oct or dB/decade or 10:1 per decade per order) ~ 1st order

• nearly 1st order but not quite. (< 1)
• 1st order slope would be A2=2 @ 4kHz instead of 5 @ 5kHz
• shape is a LPF followed by a higher HPF so 2 R ratio for A1 at f2 with break points at f1,f2 with equivalent RC
• so two 1st order filters, one LPF and one HPF , WHich order? does it matter? yes since it has a DC gain.

1= Once the low pass filter "killed" certain frequencies the high pass filter cannot "bring them back to life" (the high pass filter has constant gain for certain frequencies, let's say those higher to fh, but the low pass has ever increasing - theoretically speaking at least - attenuation for those frequencies. So the low pass filter wins).

2- The frequency response shown is easily obtained with a pole at frequency fl and a zero at frequency fh

What you're looking after is called a shelving filter.

These are the topologies and design equations that you can use for an analog implementation of it: