Even And Odd Signals

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Even functions

ƒ(x) = x² is an example of an even function.

Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all x in the domain of f:

Geometrically, the graph of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis.
Examples of even functions are |x|, x², x⁴, cos(x), and cosh(x).

 Odd functions

ƒ(x) = x³ is an example of an odd function.

Again, let f(x) be a real-valued function of a real variable. Then f is odd if the following equation holds for all x in the domain of f:

or

Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin.
Examples of odd functions are x, x³, sin(x), sinh(x), and erf(x).

 Some facts

ƒ(x) = x³ + 1 is neither even nor odd.

A function's being odd or even does not imply differentiability, or even continuity. For example, the Dirichlet function is even, but is nowhere continuous. Properties involving Fourier series, Taylor series, derivatives and so on may only be used when they can be assumed to exist.

[edit] Basic properties

• The only function which is both even and odd is the constant function which is equal to zero (i.e., f(x) = 0 for all x).
• The sum of an even and odd function is neither even nor odd, unless one of the functions is equal to zero over the given domain.
• The sum of two even functions is even, and any constant multiple of an even function is even.
• The sum of two odd functions is odd, and any constant multiple of an odd function is odd.
• The product of two even functions is an even function.
• The product of two odd functions is an even function.
• The product of an even function and an odd function is an odd function.
• The quotient of two even functions is an even function.
• The quotient of two odd functions is an even function.
• The quotient of an even function and an odd function is an odd function.
• The derivative of an even function is odd.
• The derivative of an odd function is even.
• The composition of two even functions is even, and the composition of two odd functions is odd.
• The composition of an even function and an odd function is even.
• The composition of any function with an even function is even (but not vice versa).
• The integral of an odd function from −A to +A is zero (where A is finite, and the function has no vertical asymptotes between −A and A).
• The integral of an even function from −A to +A is twice the integral from 0 to +A (where A is finite, and the function has no vertical asymptotes between −A and A).
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Periodic and Aperiodic Signal

Data can be analog or digital. The term analog data refers to information that is continuous; digital data refers to information that has discrete states. Analog data take on continuous values. Digital data take on discrete values

Topics discussed in this section:

§Analog and Digital Data

§ Analog and Digital Signals

§ Periodic and Nonperiodic Signals

Analog and Digital Signals

•Signals can be analog or digital.

•Analog signals can have an infinite number of values in a range.

Digital signals can have only a limited
number of values

In data communications, we commonly use periodic analog signals and nonperiodic digital signals.

Periodic analog signals can be classified as simple or composite. A simple periodic analog signal, a sine wave, cannot be decomposed into simpler signals. A composite

periodic analog signal is composed of multiple sine waves.

Topics discussed in this section

§Sine Wave

§ Wavelength

§

 

real and complex signal

In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity.
In the physical world, any quantity measurable through time or over space can be taken as a signal. Within a complex society, any set of human information or machine data can also be taken as a signal. Such information or machine data (for example, the dots on a screen, the ink making up text on a paper page, or the words now flowing into the reader's mind) must all be part of systems existing in the physical world – either living or non-living.
Despite the complexity of such systems, their outputs and inputs can often be represented as simple quantities measurable through time or across space. In the latter half of the 20th century, electrical engineering itself separated into several disciplines, specializing in the design and analysis of physical signals and systems, on the one hand, and in the functional behavior and conceptual structure of the complex human and machine systems, on the other. These engineering disciplines have led the way in the design, study, and implementation of systems that take advantage of signals as simple measurable quantities in order to facilitate the transmission, storage, and manipulation of information.
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REAL SIGNAL

Cuthbert Nyack

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The diagram above shows the waveform produced when one person speaks the word the "one". Horizontal axis is labelled in milliseconds. Both the amplitude and frequency change drastically during the utterance of the sound and this poses some difficulties in using the Fourier transform which assumes a stationary signal to analyse the sound.

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Above is an expanded view of the start of the sound illustrated above. Notice that it has the characteristics of a set of impulse responses of a damped oscillator.

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If the signal is broken up into several small segments so that the spectrum does not change very much during each segment, then the fourier transform may be applied to each segment. Above is an approximate sketch of the frequency spectrum within each segment. The signal starts from the front and goes towards the back. A more sophisticated application of the above proceedure is referred to as the short time fourier transform.

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In the above spectral plot, the signal starts from the back and moves towards the front.

A complex signal

Signal Analysis and Transmission

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Baseband bandwidth

A baseband bandwidth is equal to the highest frequency of a signal or system, or an upper bound on such frequencies,^[1] for example the upper cut-off frequency of a passband filter. By contrast, passband bandwidth is the difference between a highest frequency and a nonzero lowest frequency.

Baseband channel

A baseband channel or lowpass channel (or system, or network) is a communication channel that can transfer frequencies that are very near zero.^[2] Examples are serial cables and local area networks (LANs), as opposed to passband channels such as radio frequency channels and passband filtered wires of the analog telephone network. Frequency division multiplexing (FDM) allows an analog telephone wire to carry a baseband telephone call, concurrently as one or several carrier-modulated telephone calls.

Digital baseband transmission

Main article: Line code

Digital baseband transmission, also known as line coding,^[3] aims at transferring a digital bit stream over base-band channel, typically an unfiltered wire, as opposed to passband transmission, also known as carrier-modulated transmission.^[4] Passband transmission makes communication possible over a bandpass filtered channel, such as the telephone network local-loop or a band-limited wireless channel.
An unfiltered wire is intrinsically a low-pass transmission channel, while a line code is intrinsically a pulse wave signal that occupies a frequency spectrum of infinite bandwidth. According to the Nyquist theorem, error-free detection of the line code requires a channel bandwidth of at least the Nyquist rate, which is half the line code pulse rate.

Baseband transmission in Ethernet

The word "BASE" in Ethernet physical layer standards, for example 10BASE5, 100BASE-T and 1000BASE-SX, implies baseband digital transmission, i.e. that a line code and an unfiltered wire are used.
This is as opposed to 10PASS-TS Ethernet, where "PASS" implies passband transmission. Passband digital transmission requires a digital modulation scheme, often provided by modem equipment. In the 10PASS-TS case the VDSL standard is utilized, which is based on the Discrete multi-tone modulation (DMT) scheme. Other examples of passband network access technologies are wireless networks and cable modems.

Baseband signal

A baseband signal or lowpass signal is a signal that can include frequencies that are very near zero, by comparison with its highest frequency (for example, a sound waveform can be considered as a baseband signal, whereas a radio signal or any other modulated signal is not).^[5]
A signal "at baseband" is usually considered to include frequencies from near 0 Hz up to the highest frequency in the signal with significant power.
In general, signals can be described as including a whole range of different frequencies added together. In telecommunications in particular, it is often the case that those parts of the signal which are at low frequencies are 'copied' up to higher frequencies for transmission purposes, since there are few communications media that will pass low frequencies without distortion. Then, the original, low frequency components are referred to as the baseband signal. Typically, the new, high-frequency copy is referred to as the 'RF' (radio-frequency) signal. A baseband signal is a low frequency signal which when modulated is transmitted on various channels.
The concept of baseband signals is most often applied to real-valued signals, and systems that handle real-valued signals. Fourier analysis of such signals includes a negative-frequency band, but the negative-frequency information is just a mirror of the positive-frequency information, not new information. For complex-valued signals, on the other hand, the negative frequencies carry new information. In that case, the full two-sided bandwidth is generally quoted, rather than just the half measured from zero; the concept of baseband can be applied by treating the real and imaginary parts of the complex-valued signal as two different real signals.

Equivalent baseband signal

An equivalent baseband signal or equivalent lowpass signal is – in analog and digital modulation methods with constant carrier frequency (for example ASK, PSK and QAM, but not FSK) – a complex valued representation of the modulated physical signal (the so called passband signal or RF signal). The equivalent baseband signal is where I(t) is the inphase signal, Q(t) the quadrature phase signal, and j the imaginary unit. In a digital modulation method, the I(t) and Q(t) signals of each modulation symbol are evident from the constellation diagram. The frequency spectrum of this signal includes negative as well as positive frequencies. The physical passband signal corresponds to

where ω is the carrier angular frequency in rad/s.

In an equivalent baseband model of a communication system, the modulated signal is replaced by a complex valued equivalent baseband signal with carrier frequency of 0 hertz, and the RF channel is replaced by an equivalent baseband channel model where the frequency response is transferred to baseband frequencies.

Modulation

A signal at baseband is often used to modulate a higher frequency carrier wave in order that it may be transmitted via radio. Modulation results in shifting the signal up to much higher frequencies (radio frequencies, or RF) than it originally spanned. A key consequence of the usual double-sideband amplitude modulation (AM) is that, usually, the range of frequencies the signal spans (its spectral bandwidth) is doubled. Thus, the RF bandwidth of a signal (measured from the lowest frequency as opposed to 0 Hz) is usually twice its baseband bandwidth. Steps may be taken to reduce this effect, such as single-sideband modulation; the highest frequency of such signals greatly exceeds the baseband bandwidth.
Some signals can be treated as baseband or not, depending on the situation. For example, a switched analog connection in the telephone network has energy below 300 Hz and above 3400 Hz removed by bandpass filtering; since the signal has no energy very close to zero frequency, it may not be considered a baseband signal, but in the telephone systems frequency-division multiplexing hierarchy, it is usually treated as a baseband signal, by comparison with the modulated signals used for long-distance transmission. The 300 Hz lower band edge in this case is treated as "near zero", being a small fraction of the upper band edge.
The figure shows what happens with AM modulationhttp://www.smtkmr.com/