Mathematics 174 – Applied Mathematics 2 Problem Set 6 – An Analog Filter April 6, 2000

In this problem set we will study a rudimentary analog filter – the RC circuit we discussed in class. Recall that for any input signal f(x), where x represents time in this example, the voltage response of the system represented by the circuit is given by a solution v(x) = Γ(f)(x) of the first order ODE:

(1)  \frac{dv}{dx} + \frac{1}{RC} v = \frac{f}{RC}

(where R, C > 0 are constants – the resistance of the resistor and the capacitance of the capacitor in the circuit).

A. We will assume that the response to a zero input f ≡ 0 is zero. (In physical terms, this means that there is no initial energy in the system before the input is supplied.) Show that in that case the response to a general input f can be written as

(2)  v(x) = Γ(f)(x) = \frac{1}{RC} e^{\frac{−x}{RC}} ∫_{−∞}^{x} f(s)e^{\frac{s}{RC}} ds

B. Explain why this system is linear, time-invariant, and causal.

C. Rewriting integral on the right hand side in (2) as a convolution, identify the impulse response γ = Γ(δ) of the system (the response to a delta function input at x = 0).

D. What is the response of this system to the Heaviside step function input:

f(x) = \{ \begin{matrix}1 & if x ≥ 0 \\ 0 & if x < 0 \end{matrix}

E. Show that if f(x) is any input signal, and F (ω) = F[f](ω) is its Fourier transform, then the Fourier transform G of the corresponding output g = Γ(f) is given by

G(ω) = F[Γ(f)](ω) = 2πF (ω)H(ω),

where H is the Fourier transform of γ = Γ(δ). The function 2πH(ω) is called the frequency response of the system.

F. Show that the frequency response H(ω) of the RC circuit is:

H(ω) = \frac{1}{1 + iRCω}

G. The function |H(ω)| is called the magnitude response of the system. Plot this as a function of ω, taking R = 1, C = 5. Also, determine the interval of ω values for which |H(ω)| ≥ \frac{1}{\sqrt{2}}.

The system here is a realizable low-pass filter, whose “pass-band” is taken (by convention) to be your interval from part G. Of course, a good low-pass filter should have a sharper passband cut-off, and should have |H(ω)| closer to 1 on more of its pass-band. One sub-area of signal processing studies methods for constructing such filters using more complicated circuits. The main tool is the form of the frequency response H(ω).

H. Plot the functions

|H(ω)| = \frac{1}{\sqrt{1 + (5ω)^{2N}}}

for N = 2, 3, 4. These magnitude response functions correspond to a family of realizable low-pass filters called Butterworth filters. Each can be realized by an electrical circuit like the RC circuit above, but with more components.

Assignment

Individual write-ups due Thursday, April 13.