Signals & Systems-lec-1

SIGNALS & SYSTEMS LEC#: 01 Instructor Seema Ansari Copyright Seema Ansari

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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/ Copyright Seema Ansari

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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

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• 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. s1 (t ) = 5t • 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. Copyright Seema Ansari

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•

SIGNAL

DIGITAL

ANALOG

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

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Continuous time signal • Defined as a continuum of times. • Represented as a continuous variable function.

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• Natural signals: speech, ECG, EEG. • ECG: provides information to doctors about patient’s heart. • EEG: Electroencephalogram: provides info about activity of the brain.

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Discrete time signals • Defined at discrete times • The independent variable takes on only the discrete value • Represented as sequence of numbers

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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 noninteger values of n.

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• 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 Copyright time Seema Ansari 10 are Discrete 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.

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Figure 1

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Analog vs. Digital • The difference between analog and digital is similar to the difference between continuoustime 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. Copyright Seema Ansari

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Figure 2

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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. Copyright Seema Ansari

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Periodic vs. Aperiodic

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

• (b) An aperiodic signal Copyright Seema Ansari

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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).

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Causal vs. Anticausal vs. Noncausal Figure 4

(a) A causal signal

(b) An anticausal signal

(c) A noncausal signal Copyright Seema Ansari

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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). Copyright Seema Ansari

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Even vs. Odd

(a) An even signal

(b) An odd signal Copyright Seema Ansari

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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) Copyright Seema Ansari fulfills the requirement of an odd function (Figure21

Even vs. Odd Figure 6

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

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(b) Even part: e(t) = (f(t) +f(−t) Copyright Seema Ansari

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Even vs. Odd Figure 6

(c) Odd part: o(t) = (f(t) −f(−t)

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Even vs. Odd Figure 6

(d) Check: e(t) +o(t) =f(t) Copyright Seema Ansari

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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 guessedCopyright based Seemaon Ansarithe averages of 25

Figure 7

(a) Deterministic Signal

(b) Random Signal

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Right-Handed vs. LeftHanded • 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 Copyright Seema Ansari

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(a) Right-handed signal

(b) Left-handed signal Copyright Seema Ansari

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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) ≤∞ Copyright Seema Ansari

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

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