Chapter #2 Continuous Signal Waveforms "An Introduction to Linear Systems and Signal Analysis" Prof. Jack Kurzweil, Electrical Engineering Department, San Jose State University 12 fig. 2.3-2a fig. 2.3-2b fig. 2.3-2c fig. 2.3-2d The energy contained in the pulse APa(t) is: aAE2P= (2.3-3b) b. The Cosine Pulse: ()≤π=otherwise ;02at ;taCostaC (2.3-4a) The energy contained in the pulse ACa(t) is: 2aAE2C= (2.3-4b) c. The Triangular Pulse: ()≤−=otherwise ;02at ;at21taT (2.3-5a) The energy contained in the pulse ATa(t) is: 3aAE2T= (2.3-5b) d. The Sawtooth Pulse: ()≤=otherwise ;02at ;attaS (2.3-6a) The energy contained in the pulse ASa(t) is: 12aAE2S= (2.3-6b)Chapter #2 Continuous Signal Waveforms "An Introduction to Linear Systems and Signal Analysis" Prof. Jack Kurzweil, Electrical Engineering Department, San Jose State University 13 The proofs of eq 2.3-2a–d are left to the Problems. end of example Example 2.3-3 All of the functions in Example 2.3-2 are time limited and finite in magnitude so that they clearly contain a finite energy. This example demonstrates that a function that is infinite in duration may also be an energy signal. Consider the function ())t(uatAetf−= (2.3-7a) shown in fig. 2.3-3. fig. 2.3-3 The Decaying Exponential The energy in this function is: ()()a2A0dtat2eA0dtatAedttfE2222=∞−=∞−=∞∞−=∫∫∫ (2.3-7b) end of example 2.3-2 Shifting, Time Reversal, and Scaling of Energy Signals In earlier examples involving the unit step and the unit ramp we have already seen the use of the time shifting and time reversal. These examples are extended in fig 2.3-4 where (a > 0). The basic idea is that the unit step function is defined as: ()<≥=0 x0;0 x;1xu (2.3-8a) so that:Chapter #2 Continuous Signal Waveforms "An Introduction to Linear Systems and Signal Analysis" Prof. Jack Kurzweil, Electrical Engineering Department, San Jose State University 14 ()()()<<−≥≥−=−aor t 0at 0;aor t 0at ;1atu (2.3-8b) and: ()()()><−≤≥=−aor t 0at 0;aor t 0t-a ;1tau (2.3-8c) fig 2.3-4 Shifting and Reversal of SignalsChapter #2 Continuous Signal Waveforms "An Introduction to Linear Systems and Signal Analysis" Prof. Jack Kurzweil, Electrical Engineering Department, San Jose State University 15 Readers should pay particular attention to the combination of time shifting and reversal because these issues will arise again in the subsequent discussion of the operation of convolution that will take place in Chapter 3. The issue of time scaling is illustrated in fig. 2.3-5. In order to visualize the scaling process consider the rectangular pulse ()≤≤−=otherwise ;01x1 ;1xf1 (2.3-9) a function having discontinuities at x = ±1. Strictly following the functional notation, it follows that the function f1(2t) has its discontinuities at t = ± ½ and f1(t/2) has its discontinuities at t = ± 2. fig. 2.3-5 Time Scaling of Signals Consequently, we may say that f1(2t) is a contraction of f1(t) and f1(t/2) is an expansion of f1(t). In subsequent discussions of Fourier Series and Fourier Transforms we will see that contraction in time leads to expansion of signal bandwidth and expansion in time leads to contraction in signal bandwidth. In making this statement, we are calling upon the readers intuitive understanding of bandwidth based on the discussion of filtering in Chapter 1. There will be more rigorous discussions of signal bandwidth in subsequent chapters.Chapter #2 Continuous Signal Waveforms "An Introduction to Linear Systems and Signal Analysis" Prof. Jack Kurzweil, Electrical Engineering Department, San Jose State University 16 Finally, we examine the combined operations of time shifting, reversal, and scaling. Given a function f(t), the function f(at – b), a,b > 0 may be constructed by taking the following steps: 1. Let ()()[]abtafbatf −=− 2. Find the signal f1(t) = f(at); if 'a' is negative a time reversal should also be done. 3. Now shift the scaled signal f1(t) to the right by b/a in order to form f1(t – b/a). Note that if b/a is a negative number then the shift is to the left. These operations are further illustrated in the