Lecture XII: Ideal filters
Maxim Raginsky
BME 171: Signals and Systems
Duke University
October 29, 2008
Maxim Raginsky
Lecture XII: Ideal filters
This lecture
Plan for the lecture:
1
LTI systems with sinusoidal inputs
2
Analog filtering
frequency-domain description: passband, stopband
amplitude and phase response of ideal filters
time-domain description
3
Detailed example: ideal lowpass filter
4
Nonideal filters
Maxim Raginsky
Lecture XII: Ideal filters
LTI systems with sinusoidal inputs
Consider an LTI system with impulse response h(t):
x(t)
h(t)
y(t)
LTI
y(t) = x(t) ⋆ h(t) =
∞
Z
x(t − λ)h(λ)dλ
−∞
In the frequency domain we have
Y (ω) = H(ω)X(ω),
where X(ω) = F [x(t)], H(ω) = F [h(t)], Y (ω) = F [y(t)].
Consider a sinusoidal input of the form
x(t) = A cos(ω0 t + θ0 ),
where A is the amplitude, ω0 is the frequency, and θ0 is the phase.
Maxim Raginsky
Lecture XII: Ideal filters
LTI systems with sinusoidal inputs
x(t) =
X(ω) =
Y (ω) =
=
=
=
A cos(ω0 t + θ0 )
πA e−jθ0 δ(ω + ω0 ) + ejθ0 δ(ω − ω0 )
H(ω)X(ω)
πA e−jθ0 H(ω)δ(ω + ω0 ) + ejθ0 H(ω)δ(ω − ω0 )
πA e−jθ0 H(−ω0 )δ(ω + ω0 ) + ejθ0 H(ω0 )δ(ω − ω0 )
h
πA e−j(θ0 +∠H(ω0 )) |H(ω0 )|δ(ω + ω0 )
i
+ej(θ0 +∠H(ω0 )) |H(ω0 )|δ(ω − ω0 )
y(t) =
=
F −1 [Y (ω)]
A|H(ω0 )| cos(ω0 t + θ0 + ∠H(ω0 ))
Thus, the action of the LTI system with impulse response h(t) on a
sinusoid with amplitude A, frequency ω0 and phase θ0 is to transform the
amplitude as A → A|H(ω0 )| and the phase as θ0 → θ0 + ∠H(ω0 ).
Maxim Raginsky
Lecture XII: Ideal filters
LTI systems with sinusoidal inputs
Many signals encountered in practice are finite sums of sinusoids:
x(t) =
N
X
Ak cos(ωk t + θk )
k=1
The action of an LTI system with impulse response h(t) on such an input
is, by linearity, given by
y(t) =
N
X
Ak |H(ωk )| cos(ωk t + θk + ∠H(ωk ))
k=1
k
Thus, an LTI system changes the amplitude ratios A
Al and the relative
phases θk − θl among the different frequency components
k, l = 1, . . . , N :
Ak
Al
θk − θl
Ak H(ωk )
·
Al H(ωl )
→ θk − θl + ∠H(ωk ) − ∠H(ωl )
→
Maxim Raginsky
Lecture XII: Ideal filters
Analog filtering
These considerations naturally lead us to the notion of filtering:
processing of signals in order to enhance certain frequency components
and to reject certain others. For example, if a signal consists of a
low-frequency information-bearing portion and a high-frequency noise
portion, we can employ a filter to reject the high frequencies and thus
remove the noise.
We will look at four kinds of filters:
1
low-pass filters pass all frequencies in the range |ω| ≤ B, for some
B > 0 and reject all others
2
high-pass filters pass all frequencies in the range |ω| ≥ B, for some
B > 0 and reject all others
3
bandpass filters pass all frequencies in the range B1 ≤ |ω| ≤ B2
for some B1 , B2 > 0 with B1 < B2 and reject all others
4
bandstop filters pass all frequencies in the range |ω| ≤ B1 and
|ω| ≥ B2 for some B1 , B2 > 0 with B1 < B2 and reject all others
Maxim Raginsky
Lecture XII: Ideal filters
Analog filtering: frequency domain description
It is convenient to look at filters in the frequency domain. For each of
the four kinds of filters, we will specify the amplitude response |H(ω)|
and the phase response ∠H(ω).
We start with the amplitude response. For the four filters we have
defined above we have:
|HLP(ω)|
1
-B
B
0
|HHP(ω)|
1
ω
-B
lowpass
|HBP(ω)|
-B1
0
B1
ω
highpass
|HBS(ω)|
1
1
-B2
B
0
B2 ω
bandpass
Maxim Raginsky
-B2
-B1
0
B1
bandstop
Lecture XII: Ideal filters
B2 ω
Some filtering terminology
Given a filter H(ω), the set of frequencies ω such that |H(ω)| > 0 is
called the passband of the filter; the set of frequencies ω such that
|H(ω)| = 0 is called the stopband of the filter.
filter
lowpass
highpass
bandpass
bandstop
passband
|ω| ≤ B
|ω| ≥ B
B1 ≤ |ω| ≤ B2
|ω| ≤ B1 and |ω| ≥ B2
stopband
|ω| > B
|ω| < B
|ω| < B1 and |ω| > B2
B1 < |ω| < B2
If the input to a filter is a sinusoid A cos(ω0 t + θ0 ), then the amplitude of
the output will be equal to:
A, if the frequency ω0 is in the passband of the filter
0, if the frequency ω0 is in the stopband of the filter
Maxim Raginsky
Lecture XII: Ideal filters
Ideal filters
Next, we need to specify the phase response ∠H(ω) of the filter. We will
call a filter H(ω) ideal if
1, if ω is in the passband
|H(ω)| =
0, if ω is in the stopband
and
∠H(ω) =
−ωtd , if ω is in the passband
0,
if ω is in the stopband
where td > 0 is some constant.
The reason for calling such filters “ideal” will become clear shortly.
Maxim Raginsky
Lecture XII: Ideal filters
Phase response of ideal filters
∠HLP(ω)
∠HHP(ω)
Btd
Btd
-B
B
0
ω
-B
-Btd
-B1td
-B2td
ω
-Btd
lowpass
-B1 -B2
B
0
highpass
∠HBP(ω)
∠HBS(ω)
B2td
B1td
B2td
B1td
0
B1 B2 ω
bandpass
Maxim Raginsky
-B1 -B2
0
-B1td
-B2td
B1 B2 ω
bandstop
Lecture XII: Ideal filters
Ideal filters with sinusoidal inputs
Let’s see what happens when we feed a sinusoidal signal
x(t) = A cos(ω0 t + θ0 )
into an ideal filter H(ω). We have already seen that the output will be
y(t) = A|H(ω0 )| cos(ω0 t + θ + ∠H(ω0 )).
Since |H(ω)| = 1 when ω is in the passband and 0 when ω is in the
stopband, while ∠H(ω) = −ωtd when ω is in the passband and 0
otherwise, we can further write
A cos(ω0 (t − td ) + θ0 ), if ω0 is in the passband
y(t) =
0,
if ω0 is in the stopband
In other words, if the frequency of the sinusoid ω0 is in the passband of
the filter, then the output y(t) of the filter is a time-delayed version of
the input x(t):
y(t) = x(t − td ).
Maxim Raginsky
Lecture XII: Ideal filters
Ideal filters with sinusoidal inputs
This explains why we use the term “ideal:” an ideal filter does not distort
the input signal, only delays it (provided the input frequency is in the
passband).
We can generalize these results to periodic signals that can be
represented by sums of sinusoids,
x(t) =
∞
X
Ak cos(ωk t + θt ),
k=1
as well as to aperiodic signals that have a Fourier transform,
x(t) ↔ X(ω). In the latter case, it is convenient to visualize the action
of the filter in the frequency domain.
Maxim Raginsky
Lecture XII: Ideal filters
Detailed example: ideal lowpass filter
Let us consider in detail the lowpass filter whose amplitude and phase
response are given by
|HLP (ω)| = p2B (ω)
and
∠HLP (ω) = −ωtd p2B (ω),
where p2B (ω) is a rectangle of unit height and width 2B centered at
ω = 0.
We have
HLP (ω) = e−jωtd p2B (ω),
so that the impulse response has the form
B
B
(t − td )
hLP (t) = F −1 e−jωtd p2B (ω) = sinc
π
π
Note that the frequency response HLP (ω) is bandlimited, hence the
impulse response hLP (t) cannot be timelimited. This implies that an ideal
lowpass filter is acausal and therefore cannot be operated in real time.
Maxim Raginsky
Lecture XII: Ideal filters
Nonideal filters
In fact, it can be shown that any ideal filter is necessarily acausal, and
therefore cannot be operated in real time. In practice, we have to resort
to causal approximations of ideal filters. For example, an ideal lowpass
filter can be approximated by an RC filter whose frequency response is
described by
1
|HRC (ω)| = p
(ωRC)2 + 1
and
∠HRC (ω) = tan−1 (−ωRC)
1
-B
0
-B
0
Maxim Raginsky
ω
B
B
ω
Lecture XII: Ideal filters