Examples of signals:Motion. The motion of a particle through some space can be considered to be a signal, or can be represented by a signal. The domain of a motion signal is one-dimensional (time), and the range is generally three-dimensional. Position is thus a 3-vector signal; position and orientation is a 6-vector signal.Sound. Since a sound is a vibration of a medium (such as air), a sound signal associates a pressurevalue to every value of time and three space coordinates. A microphone converts sound pressure at some place to just a function of time, generating a voltage signal as an analog of the sound signal. Sound signals can be sampled to on a discrete set of time points; for example, compact discs (CDs) contain discrete signals representing sound, recorded at 44,100 samples per second; each sample contains data for a left and right channel, which may be considered to be a 2-vector signal (since CDs are recorded in stereo).Images. A picture or image consists of a brightness or color signal, a function of a two-dimensional location. A 2D image can have a continuous spatial domain, as in a traditional photograph or painting; or the image can be discretized in space, as in a raster scanned digital image. Color images are typically represented as a combination of images in three primary colors, so that the signal is vector-valued with dimension three.Videos. A video signal is a sequence of images. A point in a video is identified by its two-dimensional position and by the time at which it occurs, so a video signal has a three-dimensional domain. Analog video has one continuous domain dimension (across a scan line) and two discrete dimensions (frame and line).Biological membrane potentials. The value of the signal is a straightforward electric potential ("voltage"). The domain is more difficult to establish. Some cells or organelles have the same membrane potential throughout; neurons generally have different potentials at different points. These signals have very low energies, but are enough to make nervous systems work; they can be measured in aggregate by the techniques of electrophysiology.

A signal is any time-varying or spatial-varying quantity. Lecture: 11

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FA-18 Hornet breaking sound barrier The white halo is formed by condensed water droplets thought to result from a drop in air pressure around the aircraft

View from the Window at Le Gras (1826), Nicéphore Niépce. Generally considered the first surviving stabilized photograph of a scene from nature taken with a camera

Composite video waveform: color bars.

Complete NTSC Frame ScanParts of the Video Signal

The three main components that together form a composite video signal are as follows:1. The luma signal (or luminance) — contains the intensity (brightness or darkness) information of the video image2. The chroma signal — contains the color information of the video image3. The synchronization signal — controls the scanning of the signal on a display such as the TV screen

Cell membrane detailed diagram

Understanding the basic concepts of signals and sequences is a requirement in the development of any system that involves the processing, manipulation, and/or transmission of signals and/or sequences.

An Introduction to Signals and Sequences

Signals and sequences are basically the same. A signal is considered analog and is operated in a continuous-time system. A continuous-time system is a system that operates on and generate signals that may vary over the entire time interval, usually t ∈[0,∞) . An example of a signal is best described in Fig. 1. The figure shown in Fig. 1 is an example of a sine wave signal of the general equation y(t) = Asin ωtin which, the amplitude A is equal to 1.0, and the angular frequency ω is also equal to 0.5, making the equation:

)/..2sin()2sin()(

)sin()(

)21sin()(

si ffiyftty

wtty

tty

ππ

===

=

Fig. 1. An example of a sinusoidal signal (one of very commen used signals in Physics & Electronics) in continuous-time.

Lecture: 11

y(t): any physical quantity (Analog signal)w: angular frequency (measured in radians/sec)f: frequency (measured Hertz, cycles/sec)i: discrete sampling-points.fs: sampled frequency (samples/sec)

A sequence, on the other hand, is considered discrete in nature, and is operated in a discrete-time system. A discretetime system is a system that generally receives inputs at equally-spaced (uniform) time intervals. Fig. 2 shows an example of a sine wave sequence of an angular frequency of 0.5, making the equation

[ ]

[ ]19,021sin

∈

=

nfor

nny

Fig. 2. An example of a sinusoidal signal in discrete-time.

CONTINUOUS-TIME SIGNALSA. Unit Step FunctionA unit step function, denoted by u (t ) is a signal (or function) that has an amplitude of 1 for time equal or greater than 0 (t ≥ 0 ), and 0.0 for time less than 0 (t < 0 ). It mathematical definition is

<≥

=0,00,1

)(tt

tu

The unit step function (see Fig. 3) is one of the most important functions used by mathematicians and engineers in the analysis and design of continuous-time systems. In addition to being useful, step functions also provide a way to “turn on” and “turn off” other functions as well, due to its “on-off” characteristics. Thus, we can think of step functions as mathematical switches.

Fig. 3. The unit step function.

A unit impulse function, denoted by δ (t ) is not really a function in the normal sense, because it is generally defined as the limit of a function or through its properties. The general definition of a unit impulse function is

B. Unit Impulse Function

),(lim)(0

σσδ ∆=→∆

fwhere, f (Δ, σ ) is any function such that when Δ → 0 , the height of the function becomes infinitely large, making its area equal to 1.0. A convenient mathematical representation of unit impulse function is

∫∞

∞−

=

≠=

=

1)(

0,00,1

)(

t

whichtt

t

δ

δ

The unit impulse (Fig. 4) is also an important signal in the study of continuous-time systems because this signal can represent any arbitrary analog signal. This is because an arbitrary analog signal can be represented (or approximated) by series of impulses in the time domain [ t = τ1, τ2] .

Fig. 4. The unit impulse function.

C. Ramp FunctionThe ramp function, denoted by r(t) is a signal whose amplitude increases proportionally as time increases. Themathematical definition of a ramp signal is

<≥

=0,00,

)(ttkt

tr

Fig. 5 shows a ramp signal of the function

<≥

=0,00,2

)(ttt

tr

A discrete-time form a ramp signal is called a rampsequence and is shown in Fig. 6.

Fig. 5. The ramp function.

Fig. 6. The ramp sequence.

D. Exponential FunctionThe exponential function is a signal whose amplitude exponentially increases or exponentially decreases, depending on the value of a , as time approaches infinity. The exponential function is defined by the equation

To get an exponential function whose amplitude increases to infinity, the value of a should be a positive whole number. Fig. 7 shows an increasing exponential signal with a = 2 . To get an exponential function whose amplitude decreases to zero, the value of a should be a positive fraction which has a value less than one. Fig. 8 shows a decreasing exponential signal with a= ½. Figs. 9 and 10 presents the discrete-time equivalents of Fig. 7 and 8, respectively.

tatx =)(

Fig. 7. Example of an increasing exponential function.

Fig. 8. Example of a decreasing exponential function.

Fig. 9. Example of an increasing exponential sequence.

Fig. 10. Example of a decreasing exponential sequence.

Exponential signals (complex representation)

MATLAB example: Discrete-time Signal Representation

t = linspace(-2*pi,2*pi,10);h = stem(t,cos(t),'fill','--');%********************************************t = linspace(0,15,100);h = stem(t,sin((pi/5)*t),'fill','--');%********************************************t = linspace(0,15,10);h = stem(t,sin(((3*pi)/5)*t),'fill','--');%********************************************t = linspace(0,32,200);h = stem(t,(sin((pi/4)*t).*cos ((pi/4)*t)),'fill','--');

Using the MATLAB functions to compute and plot the following functions

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t = (0:0.001:1)'; imp= [1; zeros(99,1)]; % Impulse unit_step = ones(100,1); % Step (with 0 initial cond.) ramp_sig= t; % Ramp quad_sig=t.^2; % Quadratic sq_wave = square(4*pi*t); % Square wave with period 0.5

plot(t,quad_sig);

%Impulse Function%n=[-3:3];%x = [(n-0)==0];%stem(n,x,'ro');

%Unit step Function%n=[-3:3]; %x=[(n-0)>=0];%stem(n,x,'ro');

%Exponential Functionn=[0:10]; x=(0.9).^n; stem(n,x,'ro')

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