ECE121 - Signals Spectra

and Signal Processing

Continuous Time

Signals

Introduction

• SIGNALS AND CLASSIFICATION OF SIGNALS

• A signal is a function representing a physical quantity or variable, and typically it contains information about the behavior or nature of the phenomenon. Mathematically, a signal is represented as a function of an independent variable t. Usually t represents time. Thus, a signal is denoted by x(t).

Classification of Time

Dependent Signals

• Continuous-Time and Discrete-Time

Signals

• Analog and Digital Signals

• Real and Complex Signals

• Deterministic and Random Signals

• Even and Odd Signals

• Periodic and Nonperiodic Signals

Continuous vs. Discrete

Transformation of Continuous-

Time Signals

• Time Transformation

– Time Reversal: y(t) = x(-t)

– Time Scaling: y(t) = x(at)

– Time Shifting: y(t) = x(t-t0)

• Amplitude Transformation: y(t) =

Ax(t) + B

y(t) = x(at + b)

1. On the plot of the original signal, replace t with τ.

2. Given the transformation τ = at + b, solve for t.

3. Draw the transformed t-axis directly below the τ-axis.

4. Plot y(t) on the t-axis.

Time Transformation of a Signal

Amplitude Transformation of a Signal

y(t) = Ax(t) + B

y(t) = 3x(t) – 1

Time and Amplitude transformation

y(t) = 3x(1 – t/2) – 1

Summary of Transformation

• Time reversal x(-t)

• Time scaling x(at)

• Time shifting x(t – t0)

• Amplitude reversal -x(t)

• Amplitude scaling Ax(t)

• Amplitude shifting x(t) + B

Signal Characteristics

• Even and Odd Signals

• Periodic and Aperiodic

Even and Odd Signals

• Even Symmetry

xe(t) = xe(-t)

• Odd Symmetry

xo(t) = -xo(-t)

Properties of Even and Odd

Signals

• The sum of two even function is even

• The sum of two odd functions is odd

• The sum of an even and an odd function

is neither even nor odd.

• The product of two even functions is

even

• The product of two odd functions is even

• The product of an even function and an

odd function is odd

Even and Odd part of a

Signal

• Signal

x(t) = xe(t) + xo(t)

• Even Part

xe(t) = ½ [x(t) + x(-t)]

• Odd Part

xo(t) = ½ [x(t) – x(-t) ]

Periodic and Aperiodic

Signals • x(t) is periodic if x(t) = x(t + nT), T > 0

• Else x(t) is aperiodic

Sum of Periodic Signals

• The sum of continuous-time

periodic signals is periodic if and

only if the ratios of the periods of

the individual signals are ratios of

integers

Fundamental Period of a Periodic signal

resulting from the sum of periodic signals

1. Convert each period ratio, T0/T0i, 2 ≤ i ≤ N, to a ratio of integers, where T01 is the period of the first signal considered and T0i is the period of one of the other N-1 signals. If one or more of these ratios is not rational, then the sum of signals is not periodic.

2. Eliminate common factors from the numerator and denominator of each ratio of integers.

3. The fundamental period of the sum of the signals is T0 = k0T01, where k0 is the least common multiple of the denominators of the individual ratios of integers.

END

Next topic:

1. Common Signals in Engineering

2. Singularity Functions

3. Mathematical Functions for Signals