Signal Transformation: Time Reversal & Scaling

• Signal transformation is a key concept in signals and systems study involving modification of the independent variable.

Time Reversal

• Time reversal maps an input signal x(t) to its output signal y(t), where y(t) = x(-t).
• The output signal y(t) is created by replacing "t" with "-t" in the input signal x(t).
• Geometrically, the output signal y(t) mirrors the input signal x(t) around the vertical line t = 0.

Time Scaling

• Time scaling transforms an input signal x(t) into an output signal y(t) via y(t) = x(at), where a is a scaling constant.
• This transformation compresses/expands the signal along the time axis and may reflect it about the vertical line t = 0.
• |a| < 1 expands the signal.
• |a| > 1 compresses the signal.
• |a| = 1 leaves the signal unchanged.
• a < 0 reflects the signal.
• Time reversal becomes a special case of time scaling when a = -1.

Time Shifting

• Time shifting maps an input signal x(t) to an output signal y(t) using y(t) = x(t - b), where b is a shifting constant.
• The output signal y(t) is derived by replacing "t" with "t – b" in x(t).
• This operation geometrically shifts the signal along the time axis.
• b > 0 shifts y(t) to the right relative to x(t), delaying it in time.
• b < 0 shifts y(t) to the left relative to x(t), advancing it in time.

Combining Time Scaling and Time Shifting

• A general transformation combines time scaling, time shifting, and time reversal, mapping x(t) to y(t) as y(t) = x(at - b), with "a" as a scaling and "b" as a shifting constant.
• The output signal y(t) is formed by replacing "t" with "at – b" in the input signal x(t).
• This transformation maintains x(t)'s shape, allowing for potential expansion/compression along the time axis and/or reflection about the vertical line t = 0.
• |a| < 1 stretches the signal along the time axis.
• |a| > 1 compresses the signal.
• a < 0 reflects the signal around the vertical line t = 0.
• This transformation is equivalent to:
  • Time shifting x(t) by b, then time scaling the result by a.
  • Time scaling x(t) by a, then time shifting the result by b/a.

Amplitude Scaling

• Amplitude scaling maps an input signal x(t) to the output signal y(t) by y(t) = ax(t), where "a" represents a scaling constant.
• Geometrically, the output signal y(t) undergoes expansion/compression in amplitude and possible reflection around the time axis.
• Amplifiers constitutes a device that achieves amplitude scaling.

Amplitude Shifting

• Amplitude shifting transforms an input signal x(t) to create an output signal y(t) = x(t) + b, where b is a shifting constant.
• From a geometric point of view, amplitude shifting contributes a vertical displacement to x(t).

Combining Amplitude Scaling and Shifting

• Amplitude scaling and shifting transformations can be combined by mapping the input signal x(t) to the output signal y(t) through y(t) = ax(t) + b.
• Here, a represents a scaling constant and b represents a shifting constant.
• This transformation is the same as first scaling x(t) by a, succeeded by shifting the resulting signal by b.

Even and Odd Signals

• A signal x(t) qualifies as even if it satisfies x(t) = x(-t) for every t.
• From a geometric perspective, an even signal displays symmetry regarding the vertical line t = 0.
• Examples encompass cosine, absolute value, and square functions.
• A signal x(t) is considered odd if it meets the condition x(t) = -x(-t) for every t.
• An odd signal is antisymmetric regarding t = 0.
• An odd signal x(t) corresponds with x(0) = 0.
• Common examples are sine, signum, and cube functions (i.e., x(t) = t³).
• Any signal x(t) can be shown as the sum x(t) = xe(t) + xo(t), where xe(t) and xo(t) stand for even and odd parts.
  • xe(t) = 0.5 * [x(t) + x(-t)], and
  • xo(t) = 0.5 * [x(t) - x(-t)].
• xe(t) and xo(t) can be written as Even{x(t)} and Odd{x(t)}.
• The sum of two even signals results in an even signal.
• The sum of two odd signals is an odd signal.
• The product of two even signals is even.
• The product of two odd signals is even.
• The product of an even and odd signal is odd.

Periodic Signals

• A signal x(t) satisfies the periodic condition, i.e., x(t) = x(t+T) for every t and a certain constant T, where T > 0.
• The period of the signal is T.
• Frequency and angular frequency have the notations f and ω:
  • f = 1/T
  • ω = 2πf = 2π/T
• Signals failing to meet the periodic condition are aperiodic.
• Periodic signals such as cosine and sine functions, also meet the condition:
  • x(t) = x(t+NT)
• This value is known as the signal's fundamental period.
• If x1(t) and x2(t) have fundamental periods T₁ and T₂, then
  • y(t) = x1(t) + x2(t) means these signals are periodic if and only if T1/T2 is a rational number. Specifically the quotient of two integers.
  • If T1/T2 = q/r
    • The fundamental period of y(t) = rT₁ = qT₂
    • Note that rT₁ is the least common multiple of T₁ and T₂.
• When summing N periodic signals x1(t), x2(t)...xN(t), with periods T1, T2...TN
  • These are periodic if and only if T1/Tk is rational for k = 2, 3,...N

Support of Signals

• Signals get sorted based on intervals where their function value reads nonzero.
• A value of x(t) refers to left sided if, for some finite constant.
  • x(t) = 0 for all t > to.
• An x(t) value is right sided when, for some finite constant.
  • x(t) = 0 for every t < to.
• Signals both left and right sided are finite duration.
• Signals that are neither left nor right sided are two sided.
• A signal x(t) relates to causal if
  • x(t) = 0 for all t < 0.
• A signal value of x(t) means anticausal should
  • x(t) = 0 for every t > 0.

Signal Energy and Power

• The energy E that refers to a signal x(t), has an expression of
  • E = Integral of |x(t)|^2 dt
• Signals that have some finite energy would mean an energy signal.
• A signal’s average power P, has this form of
  • P = limit as T approached infinity of (1 / T) times the integral of |x(t)|^2 dt
• A signal x(t), including finite average power refers to the signal power.

Elementary Signals

• Some elementary signals prove more useful especially for the purposes of more beneficial ones.

Real Sinusoidal Signals

• Real sinusoids make up an important class
• A real sinusoidal signal x(t) is defined by
  • x(t) = Acos(ωt+θ)
• A, ω and θ refer to real constants.
• The signal shows periodicity if period T = 2π/ω and has a plot resembling it.

Complex Exponential Signals

• Complex exponential signals account for a set of signals that are complex exponentials.
• A complex exponential signal x(t) reads a general format of
  • x(t) = Ae^λt
• A and λ are complex which makes up some convenience representing other kind of signals.
• Complex exponential signals demonstrate a behavior as determined for their parameters for A and λ.

Real Exponential Signals

• Real exponential signals reads the consideration for special case complex exponentials.
• For cases describing a real exponential, restrict A and λ so (2.25) becomes real.
• A real exponential might show one behavior.
  • if λ > 0, the signal x(t) will exponentially grow.
  • if λ < 0, the x(t) signal decreases exponentially. Note this has more similarity as a damped exponential.
  • if λ = 0, the x(t) signals becomes a value of constant A.

Complex Sinusoidal Signals

• Regard complex sinusoid signals accounting for the special case complex exponentials.
• For each parameter listed in (2.25), we can see A referring to being complex showing purely imaginary meaning stating Re{λ} = 0.
• To describe a polar A, Cartesian λ, using
  • A = |A|e^jθ
  • λ = jω
• Show cases for w (omega) referring to real constants. By showing Euler's relation
  • x(t) states as A times e^(λt) which = |A|e^jtheta times e^(jωt) = |A|e^j(ωt + θ)
  • A equation of |A| cos(ωt + θ) + j |A| sin(ωt + θ).

General Complex Exponential Signals

• Show examples for general complex exponential signals stating where A and λ account as complex for both.
• To recap for reasons explaining those components from cartesian forms using
  • A = |A|e^j(theta)
  • λ = σ + j(omega)

Relationship Between Complex Exponential and Real Sinusoidal Signals

• A real sinusoid presents as the couple for complex sinusoids with this function.
  • A cos(ωt + θ) = (A/2) (e^j(ωt + θ)) + e^-j(ωt + θ))
  • A sin(ωt + θ) = (A / 2j) (e^j(ωt + θ)) + e^-j(ωt + θ))
• Note results derive from Euler's relation and implies a restatement about a value.*

Unit-step Function u(t)

• Used often for the systems unit function.
• The unit function states this function to describe something.
  • u(t) = 0 when t. Note the other common values 0, 1, and 1/2, show there is limited agreements that define the existing standard.
• From the description, u(t) doesn’t specify about any t.
• We can see t having a finite value, meaning does not specify.

Unit Rectangular Pulse

• The pulse means rectangular.
• The pulse function expresses by rect(t). This shows by (2.26).
• A signal means the rectangular pulse to extract. The units rectangular function shows a function of what needs extraction from a function (2.7). This shows functions like Figure 2.17(a), with a Solution. From the solution. Extract the period by wanting a time for functions to read over this interval zero to a place. Function rect(t/T) shows. As Figure 2.1 shows. Results work in Figure 2.17.

Unit Triangular Pulse

• A useful elementary function means to use the unit pulse.
• The pulse function expresses in tri(t) with the defined expression to represent it.

Cardinal Sine Function

• Show a certain function that represents a pattern or function based on the form x(t) = sin(t)/t. We define and give one function to refer by using, sinc function, a cardinal, functions can define. sinc(t) expresses sin(t)/t. Rules for a formula can show confirming a well defined amount where functions derive and imply from the origin to express them well. Cardinal sine accounts from "sinus cardinalis" to mean functions may relate or show a way to define them.

Unit-Impulse Function o(t)

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Signal Representation Using Elementary Signals

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• Function is u(t - a0 times - u(t if we have conditions where it functions like something.

Unit Rectangular Pulse

• Given to show unit functions
• Shows what we express per element. Relates to express a function
• Given to show or show. Function what use implies in what works. If a piecewise linear representation

Linear Polynomial Functions

• A piecewise functions can show relationships that it has
• Solution when segment implies the relationship per variable
• Relates if it follows what elements we mention for

Continuous-Time Systems

• An input in equation to represent