MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.u ( t ) u ( t ) t ���� � � � � ����  1 Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 21 Reading: • Class handout: Convolution • Class handout: Sinusoidal Frequency Response Continuous LTI System Time-Domain Response A continuous linear filter is a LTI dynamical system (described by an ODE with constant coefficients). We are interested in the input-output relationships and seek a method of determining the response y(t) to a given input u(t). L T I F i l t e r y ( t ) = ? i n p u t o u t p u t s y s t e m i s a t r e s t a t t = 0 The relationship is developed as follows (see the handout for a detailed explanation) • The input u(t) is approximated as a zero-order (staircase) waveform ˜uT (t) with intervals T . ������������� � �� �� ���� ���� ������ � ���� ����������������� � � �� �� ���� ���� ������ � u˜T (t)= u(nT ) for nT ≤ t< (n +1)T. • The approximation ˜uT (t) is written as a superposition of non-overlapping pulses ∞ u˜T (t)= pn(t) n=−∞ c1copyright D.Rowell 2008 2–1 T T t 0 1 / T 0 n  �  where u(nT ) nT ≤ t< (n +1)T pn(t)= 0 otherwise For example, p3(t) is shown cross-hatched in the figure above. • Each component pulse pn(t) is written in terms of a delayed unit pulse δT (t), of width T and amplitude 1/T that is: pn(t)= u(nT )δT (t − nT )T so that ∞ u˜T (t)= u(nT )δT (t − nT )T. n=−∞ • Assume that the system response to an input δT (t) is a known function, and is desig-nated hT (t) as shown below. If the system is linear and time-invariant, the response to a delayed unit pulse, occurring at time nT is simply a delayed version of the pulse response: yn(t)= hT (t − nT ) @ ( t - n T ) y ( t ) @ ( t - n T ) s y s t e m y n ( t ) 0 n T ( n + 1 ) T 0 n T ( n + 1 ) T t • The principle of superposition allows the total system response to ˜uT (t) to be written as the sum of the responses to all of the component weighted pulses: ∞ y˜T (t)= u(nT )hT (t − nT )T n=−∞ � � �� �� ���� ���� � ������������������������� � � �� �� ���� ����� � ���������������� ���� �� For causal systems the pulse response hT (t) is zero for time t< 0, and future com-ponents of the input do not contribute to the sum, so that the upper limit of the summation may be rewritten: N y˜T (t)= u(nT )hT (t − nT )T for NT ≤ t< (N +1)T. n=−∞ 2–2 u ( t ) u ( t ) t  • We now let the pulse width T become very small, and write nT = τ, T = dτ , and note that limT →0 δT (t)= δ(t). As T → 0 the summation becomes an integral and N  y(t) = lim T →0 u(nT )hT (t − nT )T n=−∞  t = u(τ)h(t − τ )dτ (1) −∞ where h(t) is defined to be the system impulse response, h(t) = lim hT (t). T →0 Equation (??) is an important integral in the study of linear systems and is known as the convolution or superposition integral. It states that the system is entirely characterized by its response to an impulse function δ(t), in the sense that the forced response to any arbitrary input u(t) may be computed from knowledge of the impulse response alone. The convolution operation is often written using the symbol ⊗: t y(t)= u(t) ⊗ h(t)= u(τ)h(t − τ )dτ. (2) −∞ Equation (??) is in the form of a linear operator, in that it transforms, or maps, an input function to an output function through a linear operation. c o n v o l u t i o n L T I F i l t e r y ( t ) = u ( t ) OX h ( t ) i n p u t h ( t ) o u t p u t The form of the integral in Eq. (??) is difficult to interpret because it contains the term h(t − τ) in which the variable of integration has been negated. The steps implicitly involved in computing the convolution integral may be demonstrated graphically below. The impulse response h(τ ) is reflected about the origin to create h(−τ), and then shifted to the right by t to form h(t − τ ). The product u(t)h(t − τ) is then evaluated and integrated to find the response. This graphical representation is useful for defining the limits necessary in the integration. For example, since for a physical system the impulse response h(t) is zero for all t< 0, the reflected and shifted impulse response h(t − τ) will be zero for all time τ>t. The upper limit in the integral is then at most t. If in addition the input u(t) is time limited, that is u(t) ≡ 0 for t<t1 and t>t2, the limits are: ⎧ t ⎪ ⎪ ⎨ u(τ)h(t − τ )dτ for t<t2 yf (t)=  t1 t2 (3) ⎪ ⎪ ⎩ u(τ )h(t − τ )dτ for t ≥ t2 t1 2–3� � ���� � � ���� � � � ����������������������� � ���� � ������������� ����� � �������� ������������ �������� �������������� � ���������� � � ����������� ��������������� �� ���� �� � ������������������� ����������������� ��������������������� � See the class handout for further details and examples. 2–4    2 Sinusoidal Response of LTI Continuous Systems Of particular interest is the response of an LTI continuous system to sinusoidal inputs of the form u(t)= A sin(Ωt + φ), where A is the amplitude, Ω is the angular frequency (rad/s), and φ is a phase angle (rad). (We note that we can also write u(t)= A sin(2πFt + φ), where F is the frequency in Hz.) We begin by noting that a sinusoid may be expressed in terms of complex exponentials through the Euler formulas: sin(Ωt)= 1  ejΩt −