Signals & Systems interaction in the Time Domain (Systems will be LTI from now on unless otherwise stated) Course Objectives Specific Course Topics: -Basic test signals and their properties -Basic system examples and their properties -Signals and systems interaction (Time Domain: Impulse Response and convolution, Frequency Domain: Frequency Response) -Applications that exploit signal & systems interaction: system id, audio effects, filtering, AM / FM radio -Signal sampling and reconstruction (time permitting) Signals & Systems interaction in the TD Goals I. Impulse Response (IR) and Convolution Formula -Definition of IR and its use for system identification -Convolution formula and its graphical interpretation II. Properties of systems from IR and convolution -Impulse response as a measure of system memory/stability -Alternative measures of memory/stability: step response III. Applications of convolution -Audio effects: reverberation -Noise removal (i.e. signal filtering or smoothing) The Impulse response When no model is available to describe a system, then we can try to use measured data in order to build one (this process is called “system identification.”) The Impulse Response When no model is available to describe a system, then we can try to use measured data in order to build one (this process is called “system identification.”) The measured data comes from input/output experiments. δ(t) h(t) x(t) h(t) y(t) € € € € € Measuring the Impulse Response (IR) of a system is one of such experiments. By definition, the IR h ( t ) of a system is its response to the unit impulse signal. € The Impulse Response When no model is available to describe a system, then we can try to use measured data in order to build one (this process is called “system identification.”) The measured data comes from input/output experiments. δ(t) h(t) x(t) h(t) y(t) € € € € € Measuring the Impulse Response (IR) of a system is one of such experiments. By definition, the IR h ( t ) of a system is its response to the unit impulse signal. In fact, when the system is LTI, the IR is all we need to know to obtain the response of€ the system to any input. The Convolution Formula Suppose we have measured the IR h ( t ) of an LTI system. Then, given an input x ( t ) we can compute the system output from zero initial conditions as € +∞ +∞ € y(t ) = ∫ h (τ ) x ( t − τ ) d τ or y(t) = ∫ h(t − τ)x(τ)dτ −∞ −∞ (Note: this is due to the sampling property of δ ( t ) and the LTI properties of the system, check out book, page 171, for more info) € € € The Convolution Formula Suppose we have measured the IR h ( t ) of an LTI system. Then, given an input x ( t ) we can compute the system output from zero initial conditions (IC) as +∞ +∞ y(t) = h(τ)x(t − τ)€dτ y(t) = h(t − τ)x(τ)dτ ∫ or ∫ € −∞ −∞ Since IC are zero, this is a pure “forced response” of the LTI system When€ the LTI system is not initially€ at rest, then the complete system response is given by the sum of a “free-body response” (*) plus the above “forced response”: +∞ y(t) = y free−body−response (t) + ∫ h(τ)x(t − τ)dτ −∞ (*) a free body response is a particular solution to the system for zero inputs € The Convolution Formula Suppose we have measured the IR h ( t ) of an LTI system. Then, given an input x ( t ) we can compute the system output from zero initial conditions (IC) as +∞ +∞ y(t) = ∫ h ( τ ) x ( t − τ ) € d τ or y(t) = ∫ h(t − τ)x(τ)dτ € −∞ −∞ Since IC are zero, this is a pure “forced response” of the LTI system. € € When the LTI system is not initially at rest, the complete system response is given by the sum of a “free-body response” plus the above “forced response”: +∞ y(t) = y free−body−response (t) + ∫ h(τ)x(t − τ)dτ −∞ For simplicity, we will assume (unless we say otherwise) that our system is always at rest when we apply an input (y free−body−response (t) = 0) € € The Convolution Formula +∞ +∞ Two equivalent formulas: y ( t ) = ∫ h ( τ ) x ( t − τ )d τ or y(t) = ∫ h(t − τ)x(τ)dτ −∞ −∞ Observe that: € - t is treated as€ a constant in the integration - τ is the integration variable - The limits of integration + ∞ , −∞ may be simplified to finite values depending on the signals h(t) x(t) € - Both integrals give the same values, so they are equivalent € € h(t) ramp(t) x(t) 7u(t) Example: suppose = = then: +∞ € € +∞ y(t) = ∫ h(τ)x(t − τ)dτ = ∫ ramp(τ)7u(t − τ)dτ −∞ −∞ +∞ t € = ∫ τ 7u(t − τ€) dτ = ( ∫ τ 7dτ)u(t) 0 0 t 2 t 2 2 € τ t 7t = 7( ∫ τdτ)u(t) = 7 u(t) = 7 − 0 u(t) = u(t) 0 2 0 2 2 € € Graphical Interpretation of Convolution Notation: From now on, we will use a * to denote the convolution-formula operation. That is: +∞ y(t) = x(t)* h(t) = ∫ h(τ)x(t − τ)dτ −∞ The Impulse Response tells us through the convolution formula how different€ the output will be from the input. Graphical Interpretation of Convolution Notation: From now on, we will use a * to denote the convolution-formula operation. That is: +∞ y(t) = x(t)* h(t) = ∫ h(τ)x(t − τ)dτ −∞ The Impulse Response tells us through the convolution formula how different the output will be from the input. € You can look at the integral as h ( t ) being a weighting function and convolution as being a weighted average of the input over the integration interval. € Graphical Interpretation of Convolution Notation: From now on, we will use a * to denote the convolution-formula operation. That is: +∞ y(t) = x(t)* h(t) = ∫ h(τ)x(t − τ)dτ −∞ The Impulse Response tells us through the convolution formula how different the output will be from the input. € You can look at the integral as h ( t ) being a weighting function and convolution as being a weighted average of the input over the integration interval. The output value y ( t ) is€ then a compromise of the memories of the input x ( t ) from the past. In other words, the values h(τ) tell how well the system remembers x ( t − τ ). € Therefore, the IR is a measure of the memory of the system. € € € Graphical Interpretation of Convolution Consider a RC low-pass filter Assume the capacitor is initially discharged (zero energy). Suppose we apply a pulse waveform at the voltage source. This leads to charging and discharging of the capacitor. −1 1 t The impulse response of the RC low-pass filter is: h ( t ) = e RC u ( t ) RC The time constant of the exponential is RC (a small value) for example, a typical value is RC = 2.5 ×10−3 s (RC)−1 = 400 € € € Graphical Interpretation of Convolution Let us compare the signals h ( τ ) and x ( t − τ ,) and the output value y ( t ). The impulse response and input signals are: € € h( ) € τ x(τ) € € Graphical Interpretation of Convolution The output signal y ( t) becomes: h(τ€) x(τ) y(t) € € € Graphical Interpretation of Convolution What is x ( t − τ ) for different values of t ? € € x(τ) x(−τ) t = 0 € € € Graphical Interpretation of Convolution What is x ( t − τ ) for different values of t ? € € x(τ) x(0.01− τ) t = 0.01 € € € Graphical Interpretation of Convolution As time t increases from t = 0 to t = 0.01 , x ( t − τ ) and h (τ ) start to overlap. The more overlap, the higher the value of the convolution integral and the more charge in the capacitor. At t = 0.01 the voltage in the capacitor is at its maximum € value. € € € € € x(0.005 − τ),h(τ) y(0.005) ≈ 3.5 € € Graphical Interpretation of Convolution As time t increases from t = 0 to t = 0.01 , x ( t − τ ) and h (τ ) start to overlap. The more overlap, the higher the value of the convolution integral and the more charge in the capacitor. At t = 0.01 the voltage in the capacitor is at its maximum € value. € € € € € x(0.01− τ),h(τ) y(0.01) ≈ 4 € € Graphical Interpretation of Convolution As time t increases from t = 0.01 to t = 0.02 , less of the non-zero part of x ( t − τ ) overlaps with the non-zero part of h ( τ ) , and the capacitor starts discharging. At t = 0.02 the voltage of the €capacitor reaches€ the minimum€ value. € € € x(0.015 − τ),h(τ) y(0.01) ≈ 0.5 € € Graphical Interpretation of Convolution As time t increases from t = 0.01 to t = 0.02 , less of the non-zero part of x ( t − τ ) overlaps with the non-zero part of h ( τ ) , and the capacitor starts discharging. At t = 0.02 the voltage of the €capacitor reaches€ the minimum€ value. € € € x(0.02 − τ),h(τ) y(0.02) ≈ 0 € € Graphical Interpretation of Convolution h(τ) € x(τ) y(t) h ( τ ) tells us how different y ( t ) will be from x(τ) In this case the output of the system is € a “rounded” version of the€ input € € € Signals & Systems Interaction in the TD Goals I. Impulse Response (IR) and Convolution Formula -Definition of IR and its use for system identification -Convolution formula and its graphical interpretation II. Properties of systems from IR and convolution -Impulse response as a measure of system memory/stability -Alternative measures of memory/stability: step response III. Applications of convolution -Audio effects: reverberation -Noise removal (i.e. signal filtering or smoothing) Impulse Response and System Memory The memory of an LTI system will clearly define the shape of the IR (how fast it decays to zero or not.) However, from the previous discussion on convolution, we also observe that the the shape of h ( t ) is what determines how much the system recalls previous input values: The larger the range of non-negative€ values of the h ( t ) for positive t , the “more memory” the system has € €The memory of the RC low-pass filter is small and related to the IR settling time Impulse Response and System Memory Definition: The settling time of a signal is the time it takes the signal to reach its steady-state value.