BME 171: Signals and Systems Duke University October 29, 2008
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
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) = In the frequency domain we have x(t λ)h(λ)dλ Y () = H()X(), where X() = F[x(t)], H() = F[h(t)], Y () = F[y(t)]. Consider a sinusoidal input of the form x(t) = Acos( 0 t + θ 0 ), where A is the amplitude, 0 is the frequency, and θ 0 is the phase.
LTI systems with sinusoidal inputs x(t) = Acos( 0 t + θ 0 ) X() = πa [ e jθ0 δ( + 0 ) + e jθ0 δ( 0 ) ] Y () = H()X() = πa [ e jθ0 H()δ( + 0 ) + e jθ0 H()δ( 0 ) ] = πa [ e jθ0 H( 0 )δ( + 0 ) + e jθ0 H( 0 )δ( 0 ) ] [ = πa e j(θ0+ H(0)) H( 0 ) δ( + 0 ) ] +e j(θ0+ H(0)) H( 0 ) δ( 0 ) y(t) = F 1 [Y ()] = A H( 0 ) cos( 0 t + θ 0 + H( 0 )) Thus, the action of the LTI system with impulse response h(t) on a sinusoid with amplitude A, frequency 0 and phase θ 0 is to transform the amplitude as A A H( 0 ) and the phase as θ 0 θ 0 + H( 0 ).
LTI systems with sinusoidal inputs Many signals encountered in practice are finite sums of sinusoids: x(t) = N A k cos( k t + θ k ) 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 A k H( k ) cos( k t + θ k + H( k )) k=1 Thus, an LTI system changes the amplitude ratios A k A l and the relative phases θ k θ l among the different frequency components k, l = 1,...,N: A k A k H( k) A l A l H( l ) θ k θ l θ k θ l + H( k ) H( l )
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 B, for some B > 0 and reject all others 2 high-pass filters pass all frequencies in the range B, for some B > 0 and reject all others 3 bandpass filters pass all frequencies in the range B 1 B 2 for some B 1, B 2 > 0 with B 1 < B 2 and reject all others 4 bandstop filters pass all frequencies in the range B 1 and B 2 for some B 1, B 2 > 0 with B 1 < B 2 and reject all others
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() and the phase response H(). We start with the amplitude response. For the four filters we have defined above we have: H LP () 1 H HP () 1 -B 0 B -B 0 B lowpass highpass H BP () 1 H BS () 1 -B 2 -B 1 0 B 1 B 2 -B 2 -B 1 0 B 1 B 2 bandpass bandstop
Some filtering terminology Given a filter H(), the set of frequencies such that H() > 0 is called the passband of the filter; the set of frequencies such that H() = 0 is called the stopband of the filter. filter passband stopband lowpass B > B highpass B < B bandpass B 1 B 2 < B 1 and > B 2 bandstop B 1 and B 2 B 1 < < B 2 If the input to a filter is a sinusoid Acos( 0 t + θ 0 ), then the amplitude of the output will be equal to: A, if the frequency 0 is in the passband of the filter 0, if the frequency 0 is in the stopband of the filter
Ideal filters Next, we need to specify the phase response H() of the filter. We will call a filter H() ideal if { 1, if is in the passband H() = 0, if is in the stopband and H() = where t d > 0 is some constant. { td, if is in the passband 0, if is in the stopband The reason for calling such filters ideal will become clear shortly.
Phase response of ideal filters H LP () H HP () -B 0 Bt d B -B 0 Bt d B -Bt d -Bt d lowpass highpass H BP () H BS () B 1 B 2 B 1 B 2 B 2 t d B 2 t d B 1 t d -B 1 -B 2 B 1 t d -B 1 -B 2 0 -B 1 t d 0 -B 1 t d -B 2 t d -B 2 t d bandpass bandstop
Ideal filters with sinusoidal inputs Let s see what happens when we feed a sinusoidal signal x(t) = Acos( 0 t + θ 0 ) into an ideal filter H(). We have already seen that the output will be y(t) = A H( 0 ) cos( 0 t + θ + H( 0 )). Since H() = 1 when is in the passband and 0 when is in the stopband, while H() = t d when is in the passband and 0 otherwise, we can further write { Acos(0 (t t y(t) = d ) + θ 0 ), if 0 is in the passband 0, if 0 is in the stopband In other words, if the frequency of the sinusoid 0 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 t d ).
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) = A k cos( k t + θ t ), k=1 as well as to aperiodic signals that have a Fourier transform, x(t) X(). In the latter case, it is convenient to visualize the action of the filter in the frequency domain.
Detailed example: ideal lowpass filter Let us consider in detail the lowpass filter whose amplitude and phase response are given by H LP () = p 2B () and H LP () = t d p 2B (), where p 2B () is a rectangle of unit height and width 2B centered at = 0. We have H LP () = e jt d p 2B (), so that the impulse response has the form h LP (t) = F 1 [ e jt d p 2B () ] = B [ ] B π sinc π (t t d) Note that the frequency response H LP () is bandlimited, hence the impulse response h LP (t) cannot be timelimited. This implies that an ideal lowpass filter is acausal and therefore cannot be operated in real time.
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 H RC () = 1 (RC)2 + 1 and H RC () = tan 1 ( RC) 1 -B 0 B -B 0 B