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A lecture note on Signals and Systems. It covers the mathematical description of signals, periodic signals, energy and power of signals, and transformations of the independent variable. The note explains the three fundamental types of signal transformations on the independent variable: time shift, time scaling, and time reversal. It also provides examples of how to apply these transformations to a signal. useful for students studying Signals and Systems or related courses.
Typology: Lecture notes
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Reading:
t
x (t
x(t) = sin(4πt)
t
x (t
x(t) = e−t
n
x [n
x[n] = n^2 − 2
n
x [n
x[n] = cos(0. 5 n)
A CT signal x(t) is periodic with period T if T is the smallest positive number such that x(t + T ) = x(t) for all t. By induction, this implies that x(t + kT ) = x(t) for all integer k and all t. Since T is smallest value that satisfies this relationship, this also implies that x(t + kT ) 6 = x(t) for some t if k is not integer. Therefore, if a signal satisfies x(t + A) = x(t) for all t, then x(t) is periodic and A is an integer multiple of its period T. For example, x(t) = cos(2t) satisfies x(t + 2π) = cos(2t + 4π) = cos(2t) for all t. But, it’s period is π (not 2 π). The following properties of periodic functions may be useful:
A DT signal x(t) is periodic with period N if N is the smallest positive integer such that x[n + N ] = x[n] for all integer n. Likewise, induction implies that x[n + kN ] = x[n] for all integer k and all integer n. By analogy, the properties of CT periodic signals described above also apply to DT periodic signals.
2 Energy and Power
Let x(t) be a continuous time signal. Suppose v(t) = x(t) volts are applied across an R ohm resistor. Then, the instantaneous current is i(t) = (^) R^1 v(t) amps and the instantaneous power dissipation is p(t) = i(t)v(t) = (^) R^1 x(t)^2 Watts. So, the total energy dissipated over the time interval t 1 ≤ t ≤ t 2 is given by
∫ (^) t 2
t 1
p(t)dt =
∫ (^) t 2
t 1
x(t)^2 dt.
and the average power over the same interval is
1 t 2 − t 1
∫ (^) t 2
t 1
p(t)dt =
R(t 2 − t 1 )
∫ (^) t 2
t 1
x(t)^2 dt.
With this simple model in mind, the standard convention is to define the energy and power of a signal as above with R = 1.
Example 2. What is the energy from t = 0 to t = 10 in x(t) = ejωt? Integrating gives
0
|x(t)|^2 dt =
0
|ejωt|^2 dt =
0
1 dt = 10.
3 Transformation of signals in CT
t 0 > 0
0
t 0 < 0
Time advance Time delay
t
x(t)
t 0 − (^2) t 0 t 0 + 2 (^) − 2 2 t 0 − 2 t 0 t 0 + 2
0 t
x(t)
− 2 2 2 − (^) a 2 − (^) a 2 a
2 a
Time compression a > 1 Time expansion a < 1
0
x(t)
− 2 2
Time reversal
t
The are many ways of transforming a CT signal into another. For instance, we can scale it, shift it in time, differentiate it, or perform a combination of these actions. Later in this
course, we will introduce the idea of transforming a signal as a system. To familiarize you with manipulating signals, in this lecture we will examine a particular type of transformation: transformation on the independent variable of signals. More formally, let us for now restrict ourselves to transformations of the form:
x(t) −→ y(t) = x(f (t))
where x(t) is the starting signal given to us, y(t) is the signal we end up with after the transformation, and f (t) is a function of t. The arrow “→” denotes the action and direction of transformation. The function f (t) can be any well-defined function, of course, but for the study of ECEN 314, we will look at the class of affine functions
f (t) = at + b
where a and b are arbitrary real constants such that a 6 = 0. The resulting transformation of x(t) into y(t) is hence called an “affine transformation on the independent variable.” All such transformations can be decomposed into just three fundamental types of signal transformations on the independent variable: time shift, time scaling, and time reversal. They involve a change of the variable t into something else:
is the dot for x(·) at each stage, some students mistakenly implement those words as (quotes indicate what they think the transformation is):
x(t)
Advance by 6 −−−−−−−−−−→ x(t + 6) Reverse −−−−−−→ x(−(t + 6))
Compress by 2 −−−−−−−−−−−→ x(2(−(t + 6)))
The incorrect version applies transformations to the entire argument of the function x(·), which results in the signal that would come from reversing the order of the transformations as shown earlier, whereas the correct version replaces the independent variable t by some function of t.
x(−(t + 6))
0
x(t + 6)
t
x(t)
− 8 − 6 − 4 − 2 2 x(−t)
4 Transformation of signals in DT
Transformations in discrete time are analogous to those in continuous time. However, there are a few subtle points to consider. For instance, can we time shift x[n] by a non-integer delay, say to x[n − 1 /2]? If we compress the signal x[n] to x[2n], do we lose half the information stored? Finally, if we expand x[n] to x[n/2], how do we “fill in the blanks?” These are some interesting questions to think about, and we will examine them further when we study sampling. It turns out that one useful method of executing a non-integer delay is by interpolating the DT samples (“connecting the dots”) into a CT signal, shifting the CT signal, then resampling to get the final shifted DT signal. A similar trick can be used for DT expansion. We will discuss the actual methods of interpolation later. For now, we will only consider time shifting y[n] = x[n − n 0 ] with integer n 0 , time scaling y[n] = x[an] for positive integer a, and time reversal y[n] = x[−n].
