Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 211Lecture #7 EGR 261 – Signals and SystemsSignals and SystemsSignalsA signal is a set of data or information. Examples include cell phone signals, television signals, voltages or currents in circuits, etc. The signals that we consider in this course are functions of the independent variable time, but our discussions would apply equally well to signals of other independent variables. Some examples of signals are shown below.Read: Ch. 1, Sect. 1-4, 6-8 in Linear Signals & Systems, 2nd Ed. by Lathi Sound segmentt [s] v(t) [V] 20 2 4 6 8 10 12 00Voltage ramp waveform2Lecture #7 EGR 261 – Signals and SystemsSystemsA system is any process that results in the transformation of signals. A system might have an input signal x(t) and an output signal y(t) as shown below.A system may be made up of physical components, such as in electric circuits or mechanical systems, or a system may be a software algorithm that modifies a signal.SystemInput signal = x(t)y(t) = Output signalContinuous-time systemSignal Energy and Signal PowerTwo useful measures of the “size” of a signal are signal energy and signal power.3Lecture #7 EGR 261 – Signals and SystemsSignal EnergySignal energy, Ex, is defined as the area under x2(t). For example, signal energy for f(t) shown below is the shaded area shown under f2(t).Ex is defined mathematically as follows:-2x-2xsignals) valued-complex(for dt x(t) Esignals) real(for (t)dt x Eor4Lecture #7 EGR 261 – Signals and SystemsSignal PowerSignal energy must be finite in order for it to be a meaningful measure of signal size. If energy is to be finite, then the signal amplitude must decay, or it is required that signal energy 0 as |t|   so that the integral will converge.When signal energy is infinite, a more meaningful measure of the size of a signal is power. Signal power, Px, is the time average of the energy. Px is defined mathematically as follows:T/2T/2-2TxT/2T/2-2Txsignals) valued-complex(for dt x(t)T1 lim Psignals) real(for (t)dt xT1 lim PorT/2T/2-2xT/2T/2-2xsignals) periodic valued-complex(for dt x(t)T1 Psignals) periodic real(for (t)dt xT1 PorFor periodic signals with period T, Px is defined as:5Lecture #7 EGR 261 – Signals and SystemsxRMST/2T/2-2RMSP Xsosignals) periodic real,(for (t)dtxT1 XRMS value of a signalSince Px is the time average (mean) of the signal amplitude squared, it is also referred to as the mean-squared value of x(t). The square root of this quantity would be the root mean-squared value of x(t) defined as:UnitsNote that the units for signal energy, Ex, and signal power, Px, are not correct dimensionally. Ex has units of V2s if x(t) is a voltage rather than joules (V2s/) and Ex has units A2s if x(t) is a current rather than joules (A2s).Similarly, Px has units of V2 or I2 rather than watts (V2/ or A2).The units for Ex and Px are dimensionally correct, however, if we think of x(t) as the voltage across a 1 resistor or i(t) as the current through a 1 resistor.So Ex and Px should perhaps be thought of as the energy or power “capability” of the signal, rather than the actual energy or power.6Lecture #7 EGR 261 – Signals and SystemsReference: Linear Signals and Systems, 2nd Edition, by Lathi.7Lecture #7 EGR 261 – Signals and SystemsEnergy signals and power signalsEnergy signals have finite energy. All energy signals decay to zero as |t| .Power signals have finite and non-zero power. All periodic signals are power signals.Quiz:Are all energy signals also power signals? No. Any signal with finite energy will have zero power. Are all power signals also energy signals? No. Any signal with non-zero power will have infinite energy. Are all signals either energy signals or power signals? No. Any infinite-duration, increasing-magnitude function will not be either. (For example, the signal x(t) =t is neither.)8Lecture #7 EGR 261 – Signals and SystemsExampleFind the energy or power (whichever is most suitable) for each signal below.tx(t)0 4 20B)tx(t)-4 0 4 20A) 109Lecture #7 EGR 261 – Signals and SystemsExample (continued)Find the energy or power (whichever is most suitable) for each signal below.tx(t)-4 0 4 20D)tx(t)-4 0 4 20C)10Lecture #7 EGR 261 – Signals and SystemsExampleFind the power and rms value of x(t) = Csin(wt - ) shown below. See the next slide for any trigonometric identities that are required.x(t)tC -CResult: For any sinusoidal waveform, RMS value = __________________11Lecture #7 EGR 261 – Signals and SystemsTrigonometric IdentitiesIf any trigonometric identities are required for homework problems, refer to section B.7-6 in Linear Signals and Systems, 2nd Edition, by Lathi. If any of the identities are needed for a test, they will be provided.12Lecture #7 EGR 261 – Signals and SystemsRMS Values of waveformsWe won’t use RMS values much in this course, but they are important in some areas such as in AC circuit analysis. • RMS values of currents and voltages are commonly used in AC circuits.• Most AC meters read RMS values rather than peak values. Example: A typical household wall outlet has 120V RMS and frequency f = 60 Hz.a)Describe this voltage as a sinusoidal functionb)Sketch the waveform and the RMS valuec)Find the period, T, and the radian frequency, w.d)Show how the RMS value is used to calculate power and compare it to an equivalent DC circuit. 0.707I 2I I 0.707V 2V VpeakpeakRMSpeakpeakRMS13Lecture #7 EGR 261 – Signals and SystemsUseful Signal OperationsThree useful signal operations arediscussed below:•Time shifting•Time scaling•Time reversal (inversion)Time shiftingConsider a signal x(t).x1(t) = x(t – T) is the same signal delayed by T seconds.x2(t) = x(t + T) is the same signal advanced by T seconds.In general, a negative shift is a shift to the right. Similarly, a positive shift is a shift to the left.Example: Given x(t) below, sketchx1(t) = x(t – 1) and x2(t) = x(t + 1).tx(t)1 2010tx2(t) = x(t + 1)1-10tx1(t) = x(t - 1)1 20 3332-1-1Replacing every t in a waveform with t – T shifts the waveform T seconds to the right.14Lecture #7 EGR 261 –