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SIGNALS & SYSTEMS
LEC#: 01Instructor
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/
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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.
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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. • 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.
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• SIGNAL
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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.
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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 non-integer 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 are
Discrete time signals.
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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 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.
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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.
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Periodic vs. Aperiodic
• Figure 3 (a) A periodic signal with period T0
• (b) An aperiodic signal
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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. NoncausalFigure 4
(a) A causal signal
(b) An anticausal signal(c) A noncausal signal
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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).
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Even vs. Odd
(b) An odd signal
(a) An even signal
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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) fulfills the requirement of an odd function (Figure 6).
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Even vs. Odd Figure 6
(a) The signal we will decompose using odd-even decomposition
(b) Even part: e(t) = (f(t) +f(−t) )
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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)
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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 guessed based on the averages of sets of signals (Figure 7).
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Figure 7
(a) Deterministic Signal
(b) Random Signal
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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
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(b) Left-handed signal
(a) Right-handed signal
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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) ≤∞
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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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