signals & systems-lec-1

Copyright Seema Ansari 1

SIGNALS & SYSTEMS

LEC#: 01Instructor

Seema Ansari

Course Introduction

• This course deals with signals, systems, and transforms, from their theoretical mathematical foundations to practical implementation in circuits .

• At the conclusion of this course, you should have a deep understanding of the mathematics and practical issues of signals in continuous and discrete time, linear time invariant systems, convolution, and Fourier transforms.

• http://cnx.org/content/m10057/latest/

2Copyright Seema Ansari

Copyright Seema Ansari 3

Concepts of a signal

• Signal: A function that conveys information, about the state or behavior of a physical system. It could be 1D; 2D; MD(multi-dimensional)

• Information is contained in a pattern of variations of some form.

• f(t) = A Sin(wt + ϴ)• Signals are represented mathematically as functions of one or

more independent variables.• The independent variable of the mathematical

representation may be either continuous or discrete.

Copyright Seema Ansari 4

• Signal: A signal is defined as any physical quantity that varies with time, space, or any other independent variable or variables.

• Mathematically a signal may be described as a function of one or more independent variables.

• E.g. • A speech signal cannot be described by such expressions.• It may be described to a high degree of accuracy as a sum

of several sinusoids of different amplitudes and frequency.

tts 5)(1

• SIGNAL

Copyright Seema Ansari 5

DIGITALANALOG

CONTINUOUS TIME

DISCRETE TIME

A /D

• A Discrete time(DT) is not a Digital signal. In DT only time is discretized, Amplitude is a continuum.• DT when passed thru A/D convertor, it becomes a Digital signal.

Copyright Seema Ansari 6

Continuous time signal

• Defined as a continuum of times.• Represented as a continuous variable

function.

Copyright Seema Ansari 7

• Natural signals: speech, ECG, EEG.• ECG: provides information to doctors about

patient’s heart.• EEG: Electroencephalogram: provides info

about activity of the brain.

Copyright Seema Ansari 8

Discrete time signals

• Defined at discrete times• The independent variable takes on only the

discrete value• Represented as sequence of numbers

Copyright Seema Ansari 9

Discrete time signals

• A Discrete time signal x(n) is a function of an independent variable that is an integer.

• The signal x(n) is not defined for non-integer values of n.

Copyright Seema Ansari 10

• Signals must be processed to facilitate extraction of information.

• Thus the development of signal processing techniques and systems is of great importance.

• Two types of Signal processing systems:a. Continuous time systems: i/p & o/p

are Continuous time signals.b. Discrete time systems: i/p & o/p are

Discrete time signals.

Signals• Signal Classifications and Properties• Continuous-Time vs. Discrete-Time• As the names suggest, this classification is determined by whether

or not the time axis (x-axis) is discrete (countable) or continuous (Figure 1).

• A continuous-time signal will contain a value for all real numbers

along the time axis. • In contrast to this, a discrete-time signal is often created by using

the sampling theorem to sample a continuous signal, so it will only have values at equally spaced intervals along the time axis.

11Copyright Seema Ansari

Figure 1

12Copyright Seema Ansari

Analog vs. Digital• The difference between analog and digital is similar to the

difference between continuous-time and discrete-time.

• In this case, however, the difference is with respect to the value of the function (y-axis) (Figure 2).

• Analog corresponds to a continuous y-axis, while digital corresponds to a discrete y-axis.

• An easy example of a digital signal is a binary sequence, where the values of the function can only be one or zero.

13Copyright Seema Ansari

Figure 2

14Copyright Seema Ansari

Periodic vs. Aperiodic

• Periodic signals repeat with some period T, while aperiodic, or nonperiodic, signals do not (Figure 3).

• We can define a periodic function through the following mathematical expression, where t can be any number and T is a positive constant:

f(t) =f(T+t)---- (1) • The fundamental period of our function, f(t) , is

the smallest value of T that still allows Equation 1 to be true.

15Copyright Seema Ansari

Periodic vs. Aperiodic

• Figure 3 (a) A periodic signal with period T0

• (b) An aperiodic signal

16Copyright Seema Ansari

Causal vs. Anticausal vs. Noncausal

• Causal signals are signals that are zero for all negative time,

• while anticausal are signals that are zero for all positive time.

• Noncausal signals are signals that have nonzero values in both positive and negative time (Figure 4).

17Copyright Seema Ansari

Causal vs. Anticausal vs. NoncausalFigure 4

(a) A causal signal

(b) An anticausal signal(c) A noncausal signal

18Copyright Seema Ansari

Even vs. Odd

• An even signal is any signal f such that f(t) =f(−t) • Even signals can be easily spotted as they are

symmetric around the vertical axis.

• odd signal, on the other hand, is a signal f such that f(t) =−(f(−t) ) (Figure 5).

19Copyright Seema Ansari

Even vs. Odd

(b) An odd signal

(a) An even signal

20Copyright Seema Ansari

Even vs. Odd• Using the definitions of even and odd signals, we can show

that any signal can be written as a combination of an even and odd signal.

• That is, every signal has an odd-even decomposition. To demonstrate this, we have to look no further than a single equation.

f(t) = 1/2 (f(t) +f(−t) ) + 1/2 (f(t) −f(−t) )…… (2) • By multiplying and adding this expression out, it can be shown

to be true.

• Also, it can be shown that f(t) +f(−t) fulfills the requirement of an even function, while f(t) −f(−t) fulfills the requirement of an odd function (Figure 6).

21Copyright Seema Ansari

Even vs. Odd Figure 6

(a) The signal we will decompose using odd-even decomposition

(b) Even part: e(t) = (f(t) +f(−t) )

22Copyright Seema Ansari

Even vs. Odd Figure 6

(c) Odd part: o(t) = (f(t) −f(−t)

23Copyright Seema Ansari

Even vs. Odd Figure 6

(d) Check: e(t) +o(t) =f(t)

24Copyright Seema Ansari

Deterministic vs. Random

• A deterministic signal is a signal in which each value of the signal is fixed and can be determined by a mathematical expression. Because of this the future values of the signal can be calculated from past values with complete confidence.

• On the other hand, a random signal has a lot of uncertainty about its behavior. The future values of a random signal cannot be accurately predicted and can usually only be guessed based on the averages of sets of signals (Figure 7).

25Copyright Seema Ansari

Figure 7

(a) Deterministic Signal

(b) Random Signal

26Copyright Seema Ansari

Right-Handed vs. Left-Handed

• A right-handed signal and left-handed signal are those signals whose value is zero between a given variable and positive or negative infinity.

• See (Figure 8) for an example. • Both figures "begin" at t1 and then extends to

positive or negative infinity with mainly nonzero values.

• Figure 8 (a) Right-handed signal (b) Left-handed signal

27Copyright Seema Ansari

(b) Left-handed signal

(a) Right-handed signal

28Copyright Seema Ansari

Finite vs. Infinite Length

• Signals can be characterized as to whether they have a finite or infinite length set of values.

• Most finite length signals are used when dealing with discrete-time signals or a given sequence of values.

• Mathematically speaking, f(t) is a finite-length signal if it is nonzero over a finite interval.

• An example can be seen in Figure 9. Similarly, an infinite-length signal, f(t) , is defined as nonzero over all real numbers: -∞≤f(t) ≤∞

29Copyright Seema Ansari

Figure 9: Finite-Length Signal. Note that it only has nonzero values on a set, finite interval.

30Copyright Seema Ansari