Convolution and Filtering 335

5.7 Convolution and Filtering 335

the output of the filter is the signal

x N ( t) =F −1 [X()H( j)]

or the inverse Fourier transform of X() multiplied by a low-pass filter with an ideal magnitude response of 1 for − c << c where the cut-off frequency  c is chosen so that N 0 < c < ( N + 1) 0 . As such, x N ( t) is the convolution

x N ( t) = [x ∗ h](t)

where h(t) is the inverse Fourier transform of H( j), or a sinc signal of infinite support. The con- volution around the discontinuities of x(t) causes ringing before and after them, and this ringing appears independent of the value of N.

■ Example 5.16

Obtain different filters from an RLC circuit (Figure 5.9) by choosing different outputs. Let the input

be a voltage source with Laplace transform V i ( s). For simplicity, let R = 1 , L = 1 H, and C = 1 F, and assume the initial conditions to be zero.

Solution

Low-pass filter: Let the output be the voltage across the capacitor; by voltage division we have that

V i ( s)/s

V i ( s)

V C ( s) =

1 + s + 1/s = s 2 +s+1

so that the transfer function is

V C ( s)

H lp ( s) =

V ( s) i = s 2 +s+1

This is the transfer function of a second-order low-pass filter. If the input is a dc source, so that its frequency is  = 0, the inductor is a short circuit (its impedance would be 0) and

FIGURE 5.9 RLC circuit for implementing different −

filters.

C H A P T E R 5: Frequency Analysis: The Fourier Transform

the capacitor is an open circuit (its impedance would be infinite), so that the voltage in the capacitor is equal to the voltage in the source. On the other hand, if the frequency of the input source is very high, then the inductor is an open circuit and the capacitor a short circuit (its impedance is zero) so that the capacitor voltage is zero. This is a low-pass filter. Notice that this filter has no finite zeros, and complex conjugate poles.

High-pass filter: Suppose then that we let the output be the voltage across the inductor. Then again by voltage division the transfer function

V L ( s)

H hp ( s) =

V ( s) i = s 2 +s+1

is that of a high-pass filter. Indeed, for a dc input (frequency zero) the impedance in the induc- tor is zero, so that the inductor voltage is zero, and for very high frequency the impedance of the inductor is very large so that it can be considered open circuit and the voltage in the inductor equals that of the source. This filter has the same poles of the low-pass filter (this is determined by the overall impedance of the circuit, which has not changed) and double zeros at zero. It is these zeros that make the frequency response for low frequencies be close to zero.

Band-pass filter: Letting the output be the voltage across the resistor, its transfer function is

V R ( s)

H bp ( s) =

V i ( s) = s 2 +s+1

or the transfer function of a band-pass filter. For zero frequency, the capacitor is an open cir- cuit so the current is zero and the voltage across the resistor is zero. Similarly, for very high frequency the impedance of the inductor is very large, or an open circuit, making the voltage across the resistor zero because again the current is zero. For some middle frequency the serial combination of the inductor and the capacitor resonates and will have zero impedance. At the resonance frequency, the current achieves its largest value and the voltage across the resistor does too. This behavior is that of a band-pass filter. This filter again has the same poles as the other two, but only one zero at zero.

Band-stop filter: Finally, suppose we consider as output the voltage across the connection of the inductor and the capacitor. At low and high frequencies, the impedance of the LC connection is very high, or open circuit, and so the output voltage is the input voltage. At the resonance frequency  r = 1 the impedance of the LC connection is zero, so the output voltage is zero. The resulting filter is a band-stop filter with the transfer function

s 2 +1

H bs ( s) = s 2 +s+1

Second-order filters can then be easily identified by the numerator of their transfer functions. Second-order low-pass filters have no zeros, and the numerator is N(s) = 1; band-pass filters have a zero at s = 0 so N(s) = s, and so on. We will see next that such a behavior can be easily seen from a geometric approach.