Lecture #8 EGR 261 – Signals and Systems

1. Lecture #8 EGR 261 – Signals and Systems Read: Ch. 1, Sect. 1-4, 6-8 in Linear Signals & Systems, 2nd Ed. by Lathi • Classification of Signals • Several classes of signals are considered in this course: • Continuous-time and discrete-time signals • Analog and digital signals • Periodic and aperiodic signals • Energy and power signals • Deterministic and probabilistic signals Continuous-time and discrete-time signals Continuous-time signals are signals that are specified for a continuum of time values (i.e., all values of time over a specified range). Discrete-time signals are signals that are only defined at discrete times (i.e., for a specific set of time values.) For example, a discrete-time signal may have values defined once per millisecond. A discrete-time signal might be a set of data point measured at specific time intervals, such as annual population figures. A discrete-time signal might also be formed by “sampling” a continuous-time signal (measuring its value at specific time intervals to produce a sequence of data points).

2. x(t) C t q x[n] -C x[3] x[5] x[2] x[4] x[6] x[9] x[1] x[0] x[7] x[8] n 0 1 2 3 4 5 6 7 8 9 Lecture #8 EGR 261 – Signals and Systems Example: Continuous-time signal x(t) is defined for any value of t. In this case a function x(t) = Csin(wt - ) describes the signal. Example: Discrete-time signal A series of data points (x[0], x[1], x[2],…) represents the signal.

3. Lecture #8 EGR 261 – Signals and Systems Analog and Digital Signals Analog signals are signals that can have any amplitude. Digital signals are signals that can only have specific amplitudes (such as binary signals that can only have values 0 or 1). Analog and digital signals are sometimes confused with continuous-time and discrete-time signals. The difference can be summarized by the following: • Continuous-time signals can have any value of time (any x value). • Discrete-time signals can have only specific values of time (only a set of x values). • Analog signals can have any amplitude (any y value). • Digital signals can have only specific amplitudes (only a set of y values).

4. Lecture #8 EGR 261 – Signals and Systems Reference: Linear Signals and Systems, 2nd Edition, by Lathi.

5. Example: x(t) is a periodic waveform x(t) Note that for any value t, such as t1, the waveform has the same amplitude. So x(t1) = x(t1 + To) t t1 t1 + To Example: Is x(t) is a periodic waveform? If so, what is the period To? y(t) t -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Lecture #8 EGR 261 – Signals and Systems Periodic and Aperiodic Signals A signal x(t) is said to be periodic if for some positive constant To x(t) = x(t + To) for all t The smallest value of To that satisfies the equation above is the fundamental period of x(t). If a signal is not periodic, then it is aperiodic.

6. x1(t) x3(t) t t x2(t) x4(t) t t Lecture #8 EGR 261 – Signals and Systems • Notes on periodic signals • A periodic signal by definition: • Remains unchanged when time-shifted by N periods (where N is an integer) • Is an everlasting signal (exists over the range - < t < ) • The area under the curve for a periodic signal is the same for any interval of duration To Causal, noncausal, and anticausal signals A signal that does not start before t = 0 is a causal signal. A signal that starts before t = 0 is a noncausal signal. A signal that is zero for all t > 0 is an anticausal signal. Note that all periodic signals are non-causal. Examples: Identify each signal below as causal, noncausal, or anticausal.

7. Lecture #8 EGR 261 – Signals and Systems • Energy signals and power signals • Earlier we discussed how to calculate signal energy and signal power. • To summarize: • A signal with finite energy is an energy signal. • A signal with finite and nonzero power is a power signal. • A signal cannot be both an energy signal and a power signal since: • A signal with finite energy has zero power. • A signal with finite power has infinite energy. • All periodic signals are power signals. • Some signals are neither energy signals nor power signals. • Examples: x(t) = t, x(t) = KtN (for N > 1), x(t) = Ke-at (defined for all time) Deterministic signals and random signals Signals that can be completely described mathematically or graphically are deterministic signals. Signals that cannot be predicted precisely, but are known only in terms of probabilistic description (such as mean value), are random signals. Examples of random signals include noise, atmospheric disturbances, stock market values, etc. This course will only deal with deterministic signals.

8. Lecture #8 EGR 261 – Signals and Systems • Useful signal models • Three important functions are used commonly in the area of signals and systems: • Unit step function, u(t) • Impulse function, (t) • Exponential function, est Unit step function, u(t) The unit step function should be familiar from EGR 260 – Circuit Analysis. A brief summary is shown below. Definition: u(t) = unit step function where and u(t) is represented by the graph shown below.

9. x(t) 6 t 0 4 2 Lecture #8 EGR 261 – Signals and Systems • Two useful skills using unit step functions: • 1) Determining the function that represents a given graph • Approach: Represent each unique portion of the function using unit step “windows” • 2) Graphing a function specified by unit steps • Approach: As each unit step function “turns on”, graph the cumulative function. Example: Represent x(t) shown below using unit step functions. Example: Graph the function x(t) = 2tu(t) + (4-2t)u(t-2) + (8-2t)u(t-4) + (2t-12)u(t-6)

10. Lecture #8 EGR 261 – Signals and Systems Impulse function, (t) Recall that the impulse function was defined earlier (and is repeated here). (t) = impulse function (also called the Dirac delta function) The impulse function is defined as: Graphically (t) is shown as: Illustration To illustrate the concept that the area under (t) = 1 (not the height =1), consider the function f(t):

11. d (t) (0)(t) Area (or strength) = (0) Area (or strength) = 1 d (t - T) (T)(t - T) Area (or strength) = (T) Area (or strength) = 1 t t 0 0 t t T T 0 0 Lecture #8 EGR 261 – Signals and Systems Important relationships related to the impulse function If a function (t) is continuous at t = 0 and since (t) = 0 for t  0, then (t)(t) = (0)(t) So the result is an impulse of strength (0). Illustration: Similarly, since (t - T) = 0 for t  T, then (t)(t - T) = (T)(t - T) So the result is an impulse of strength (T). Illustration:

12. Lecture #8 EGR 261 – Signals and Systems Sifting Property Since we have just seen that (t)(t) = (0)(t) it follows that (Sifting Property) (Note that we will see more of this useful property) Similarly Proof:

13. Lecture #8 EGR 261 – Signals and Systems Example: Evaluate each integral below Relationship between u(t) and (t) We can use integration by parts to show that But the sifting property states that This yields the following important result: or

14. Lecture #8 EGR 261 – Signals and Systems Exponential function, est Recall that s = complex frequency where s =  + jw so est = e( + jw)t = etejwt = et[cos(wt) + jsin(wt)] (Eq. 1) Also note that since s* =  - jw (the complex conjugate of s) then es*t = e( - jw)t = ete-jwt = et[cos(wt) - jsin(wt)] (Eq. 2) Combining Eq. 1 and Eq. 2 above yields: etcos(wt) = ½[est + es*t]

15. Lecture #8 EGR 261 – Signals and Systems • Functions represented by est • Note that est represents four types of functions (show the form of each and sketch): • Constants • Exponential functions • Sinusoids • Exponentially varying sinusoids