8 Linear System Theory 8.1 INTRODUCTION In this chapter, the fundamental relationships between the input and output of a linear time invariant system, as illustrated in Figure 8.1, are detailed. Specifically, the relationships between the input and output time signals, Fourier transforms and power spectral densities, are established. Such relation- ships are fundamental to many aspects of system theory, including analysis of noise in linear systems, and low noise amplifier design. The relationships between the parameters defined in Figure 8.1, and proved in this chapter are, y(t) :  R  x()h(t 9 ) d (8.1) Y (T, f ) H(T, f )X(T, f ) (8.2) where X and Y are the respective Fourier transforms, evaluated on the interval [0, T ], of the signals x and y. However, as will be shown in this chapter, the relationship defined in Eq. (8.2) is an approximation. If both x, h + L , then the relative error in this approximation can be made arbitrarily small by making T sufficiently large. However, stationary random signals are not Lebesgue integrable on the interval (0, -) and hence, this convergence is not guaranteed. However, it is shown, for a broad class of signals and random processes, including periodic signals and stationary random processes, that the corre- sponding relationship between the input and output power spectral densities, namely, G 7 (T, f ) "H(T, f )"G 6 (T, f ) (8.3) becomes exact as T increases without bound. Establishing the relationships, as per Eqs. (8.1)—(8.3), for a linear time invariant system requires the system impulse response to be defined, and this is the subject of the next section. 229 Principles of Random Signal Analysis and Low Noise Design: The Power Spectral Density and Its Applications. Roy M. Howard Copyright ¶ 2002 John Wiley & Sons, Inc. ISBN: 0-471-22617-3 x ∈ E X G X G Y h ↔ H y ∈ E Y Figure 8.1 Schematic diagram of a linear system. E X and E Y , respectively, represent the ensemble of input and output signals. H is the Fourier transform of the impulse response function h. G X and G Y , respectively, are the power spectral densities of the input and output random processes. ∆ t ∞ ∫ –∞ δ ∆ (t)dt = 1 δ ∆ (t) 1 ∆ − Figure 8.2 Definition of the function   . 8.2 IMPULSE RESPONSE Fundamental to defining the impulse function of a time invariant linear system, is the function   defined by the graph shown in Figure 8.2. The response of a linear time invariant system to the input signal   is denoted h  . D:I R By definition, the impulse response of a linear system is the output signal, in response to the input signal   ,as becomes increasingly small, that is, h(t) : lim  h  (t) (8.4) 8.2.1 Restrictions on Impulse Response General requirements on the impulse function are, first, that it is integrable, that is, h + L [0, -], and second, that as  ; 0, the integrated difference between h and h  is negligible on sets of nonzero measure, that is, convergence in the mean over (0, -): lim     "h  (t) 9 h(t)"dt : 0 (8.5) 230 LINEAR SYSTEM THEORY t i i β i − 1 i + 1 i + i 1 + 3β ⁄ 2 1 Figure 8.3 Illustration of a function that is Lebesgue integrable on [0, -], but is not square Lebesgue integrable on the same interval. The following are two examples where, as  ; 0, the integrated error between h and h  is finite. First, the ‘‘identity’’ system where h  (t) :   (t) and second, the system where h  (t) :   0 t+ [,  ; 1/] elsewhere (8.6) For both systems, and for t + (0, -), it follows that h(t) : lim  h  (t) : 0 but lim     "h  (t) 9 h(t)"dt " 0 (8.7) To ensure h + L [0, -], and as  ; 0 the integrated difference between h and h  is negligible, the following restriction on the set of functions +h  ,, denoted condition 1, is sufficient: 1. There exists a function g + L [0, -], such that, for all 90 it is the case that "h  (t)"-g(t) for t + [0, -]. The validity of this condition, in terms of guaranteeing that Eq. (8.5) holds, is given by Theorem 2.25. Practical and stable systems are such that h  is bounded and has finite energy for all values of . As per Theorem 2.14 these two criteria are met by condition 1 and the following condition. 2. For 90, h  is bounded, that is, 90, "h  (t)"-h  for t+ [0, -]. This second condition excludes a signal such as 1/(t , which is integrable on [0, -], but has infinite energy on all intervals of the form [0, t M ]. It also excludes signals such as the one shown in Figure 8.3, whose integral equals   G (1/i>@), which from the comparison test (Knopp, 1956 pp. 56f ), is finite for 90, but whose energy is given by   G (1/i\@) and is infinite when 90. IMPULSE RESPONSE 231 8.3 INPUT ‒OUTPUT RELATIONSHIP Consider the causal linear time invariant system illustrated in Figure 8.1. The well-known relationship between the input and output signals is specified in the following theorem. T 8.1. I—O R   L S If the input signal x to, and the system impulse response h of, a linear time invariant system are both causal, are locally integrable, and have bounded variation on all finite intervals, then the output signal, y, is given by y(t) :  R  x()h(t 9 ) d t 9 0 (8.8) Proof. The proof of this result is given in Appendix 1. Note that this result is applicable to unstable systems where h, L [0, -]. 8.4 FOURIER AND LAPLACE TRANSFORM OF OUTPUT The following theorem states the important result of the relationship between the Fourier and Laplace transforms of the input and output of a linear time invariant system. T 8.2. T  O S   L S If both x,h+ L [0, T ], have bounded variation on [0, T ], and their respective Fourier transforms are denoted X and H, then the Fourier transform Y of the output signal y, evaluated on [0, T ], is given by Y (T, f ) :  WNH2 N>HW2 x()h(p)e\HLDN>H d dp (8.9) : Y  (T, f ) 9 I(T, f ) : X(T, f )H(T, f ) 9 I(T, f ) where Y  (T, f ) : X(T, f )H(T, f ) (8.10) I(T, f ) :  WNH2 N>H2 x()h(p)e\HLDN>H d dp 232 LINEAR SYSTEM THEORY Figure 8.4 Illustration of area of integration for Y and I. and the integration regions for both Y and I are as shown in Figure 8.4. W ith X(T, s) :  2  x(t)e\QR dt (8.11) and similarly for other L aplace transformed variables, it is the case that Y (T, s) :  WNH2 N>HW2 x()h(p)e\QN>H d dp (8.12) : X(T, s)H(T, s) 9 I(T, s) where I(T, s) :  WNH2 N>H2 x()h(p)e\QN>H d dp (8.13) Proof. The proof of this theorem is given in Appendix 2. For the Fourier transform case Y  (T, f ), because of its simplicity, is the approximation that is normally used, and I(T, f ) is clearly the error between the approximate and true output Fourier transforms for a given frequency f. The next theorem gives a sufficient condition for the term I to approach zero as the interval under consideration becomes increasingly large. T 8.3 C  A If both x, h + L [0, -], and have bounded variation on all closed finite intervals, then lim 2 Y (T, f ) : lim 2 X(T, f )H(T, f ) f + R (8.14) lim 2 Y (T, s) : lim 2 X(T, s)H(T, s) Re[s] . 0 (8.15) FOURIER AND LAPLACE TRANSFORM OF OUTPUT 233 t T t T t T x(t) h(t) y(t) 2T Figure 8.5 Illustration of waveforms in a linear system for the case where the impulse response and the input are windowed but the output is not. Further, if h + L [0, -], x is locally integrable and does not exhibit exponential increase, then Re[s] 9 0 is a sufficient condition for lim 2 Y (T, s) : lim 2 X(T, s)H(T, s) (8.16) Proof. The proof is given in Appendix 3. 8.4.1 Windowed Input and Nonwindowed Output For completeness, the response of a linear time invariant system for the case where the input and impulse response are windowed, but the output is not windowed, as illustrated in Figure 8.5, is stated in the following theorem. T 8.4 T  O S:N C If both x, h + L [0, T ], and have bounded variation on [0, T ], then the Fourier and L aplace transforms Y  of the output signal y, which is not windowed, are given by Y  (2T, f ) : X(T, f )H(T, f ) (8.17) Y  (2T, s) : X(T, s)H(T, s)(8.18) Proof. The proof of this result is given in Appendix 4. This result has application, when the output signal y is to be derived for the interval [0, T ]. The procedure is as follows for the Fourier transform case. First, evaluate X(T, f ) and H(T, f ), second, evaluate Y  (2T, f ):X(T, f )H(T, f ), and third, evaluate y by taking the inverse Fourier transform of Y  (2T, f ). The evaluated response is valid for the interval [0, T ], but not [T,2T ]. 8.4.2 Fourier Transform of Output — Power Input Signals Theorem 8.3 states that lim 2 Y  (T, f ) : lim 2 Y (T, f ), provided x, h+ L. However, for the common case of signals whose average power evaluated on [0, T ], does not significantly vary with T, for example, stationary or periodic 234 LINEAR SYSTEM THEORY Time (Sec) 0.25 0.5 0.75 1 1.25 1.5 1.75 2 y(t) −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Figure 8.6 Output waveform y. signals, it is the case that x , L . For this situation, it can be the case that lim 2 Y  (T, f ) " lim 2 Y (T, f ) almost everywhere. The following example illustrates this point. 8.4.2.1 Example Consider a linear system with an impulse response and input signal, respectively, defined according to h(t) : h M e\RO  t 9 0, 90 x(t) :(2  V sin(2f V t) t 9 0 (8.19) For the case where  V : 1, h M : 1,  : 0.1, T : 1, and f V : 4, the output signal y is plotted in Figure 8.6. For these parameters, the magnitude of the true, Y, and approximate, Y  , Fourier transforms, as well as the magnitude of the error, I, between these transforms, is plotted in Figure 8.7. To establish bounds on the integral I, and hence, on how well Y  approxi- mates Y, note that "I(T, f )" :  2   2 2\H h(p)e\HLDN dp  x()e\HLDH d  -(2  V  2   2 2\H h M e\NO  dp  d : (2  V h M   19e\2O 9 Te\2O   (8.20) When T is sufficiently large, such that Te\2O/ 1, it follows that "I(T, f )"- (2  V h M , which is independent of the interval length T, and only depends on FOURIER AND LAPLACE TRANSFORM OF OUTPUT 235 0.5 1 5 10 50 0.005 0.01 0.05 0.1 0.5 True Frequency (Hz) Magnitude Approximation Error Figure 8.7 Magnitude of the true and approximate Fourier transform of the output signal as well as the magnitude of the error between these two transforms, for the case where T : 1. 3 4 5 6 7 8 0.01 0.02 0.05 0.1 0.2 0.5 1 2 Magnitude Frequency (Hz) Figure 8.8 Magnitude of the true Fourier transform of the output signal for the cases where T : 2 (lower peak) and T : 4 (higher peak). the input signal amplitude and the system impulse response characteristics h M and . For the given parameters, the bound for "I(T, f )" is 0.141. From Figure 8.7, it follows that the maximum magnitude of I is 0.05, which is within this bound. Further, the level of the error defined by "I" does not increase or decrease as the interval length T increases (see Figures 8.8 and 8.9). In Figure 8.8 the 236 LINEAR SYSTEM THEORY 3 4 5 6 7 8 0.01 0.02 0.05 0.1 0.2 0.5 1 2 Frequency (Hz) Magnitude error Figure 8.9 Magnitude of the approximate Fourier transform Y  of the output signal, for the cases where T : 2 (lower peak) and T : 4 (higher peak). The magnitude of the error between the true and approximate Fourier transform is identical for T : 2 and T : 4, and is the smooth curve. magnitude of the true Fourier transform Y, is plotted for cases T : 2 and T : 4. In Figure 8.9, the magnitude of approximate Fourier transform Y  ,as well as the error "I", are graphed for cases T : 2 and T : 4. As T increases, the lobe at the frequency of the input (4Hz) increases in height, and decreases in width. Away from the lobe, the envelope of the magnitude of both Y and Y  remains constant as T increases and, consistent with this, I does not change with T. Clearly, for this example the approximate Fourier transform Y  , does not converge to the true Fourier transform Y, defined in Eq. (8.9). 8.4.2.2 Explanation An explanation of the nonconvergence of Y  (T, f )to Y (T, f )asT ; -, for signals with constant average power, can be found by noting that I can be approximated by an integral over the region defined in Figure 8.10, where t F is a time such that   R F "h(p)" dp    "h(p)" dp. The magni- tude of this integral is relatively insensitive to an increase in the value of T. That is, as T increases the error defined by "I" remains relatively static. For the case where x + L , the magnitude of  2 2\R F "x()" d decreases, in general, as T increases, and the error defined by "I" converges to zero. 8.4.2.3 Power Spectral Density Clearly, on a finite interval [0, T ], it is the case that G 7 (T, f ) : "Y (T, f )" T " "X(T, f )""H(T, f )" T a.e. (8.21) FOURIER AND LAPLACE TRANSFORM OF OUTPUT 237 λ T T p to I p = T −λ t h Region where there is significant contribution T – t h Figure 8.10 Region of integration where there is a significant contribution to the integral I. The time t h is the time when the impulse response has negligible magnitude as defined in the text. as I(T, f ) is finite. However, for the infinite interval, it follows, as I(T, f ) does not increase with T, that lim 2 G ' (T, f ) : lim 2 "I(T, f )" T : 0 (8.22) A consequence of this result is that lim 2 G 7 (T, f ) : lim 2 "X(T, f )""H(T, f )" T : lim 2 G 6 (T, f )"H(T, f )" (8.23) In fact, as shown in the next section, this last result holds for a broad class of signals that are not elements of L [0, -]. 8.5 INPUT — OUTPUT POWER SPECTRAL DENSITY RELATIONSHIP Consider the case where the input random process X to a linear system, is defined on the interval [0, T ] by the ensemble E 6 :  x: S  ;[0, T ] ; R S  3 Z> countable case S  3 R uncountable case  (8.24) where P[x(, t)] : P[] : p A for the countable case, and P[x(, t)" AZA M A M >BA ] : P[+ [ M ,  M ; d]] : f  ( M ) d for the uncountable case. Here, f  is the prob- ability density function associated with the index random variable , whose sample space is S  . 238 LINEAR SYSTEM THEORY