SIGNAL PROCESSING

-Bhaumik Vaidya

Signals

An electric signal is a voltage or current waveform whose time or frequency variations correspond to the desired information.

The information-bearing signals are processed either for purposes of measurement in an instrumentation system, or for transmitting over long distance in a communication system.

Continuous signals  are described by time functions which are defined for all values of t (a continuous variable). Commercial broadcast systems, analog computers, and various control and instrumentation systems process continuous signals.

Discrete signals on the other hand, exist only at specific instances of time, and as such, their functional description is valid only for discrete-time intervals.  Digital computers, pulsed-communication systems (modern telephone and radar), and microprocessor-based control systems utilize discrete signals.

Signal Processing System

Figure  shows the functional block diagram of a signal-processing system. The information source may be a speech (voice), an image (picture), or plain text in some language.

A transducer is usually required to convert the output of a source into an electrical signal that is suitable for transmission. Typical examples include a microphone converting an acoustic speech or a video camera converting an image into electric signals. A similar transducer is needed at the destination to convert the received electric signals into a form (such as voice, image, etc.)

The heart of any communication system consists of three basic elements: transmitter, transmission medium or channel, and receiver. The transmitter (input processor) converts the electric signal into a form that is suitable for transmission through the physical channel or transmission medium.

For example, in radio and TV broadcasts, since the FCC (Federal Communications Commission) specifies the frequency range for each transmitting station, the transmitter must translate the information signal to be transmitted into the appropriate frequency range that matches the frequency allocation assigned to the transmitter. This process is called modulation, which usually involves the use of the information signal to vary systematically the amplitude, frequency, or phase of a sinusoidal carrier.

Thus, in general, carrier modulation such as amplitude modulation (AM), frequency modulation (FM), or phase modulation (PM) is performed primarily at the transmitter. For example, for a radio station found at a setting of AM820, the carrier wave transmitted by the radio station is at the frequency of 820 kHz.

The function of the receiver is to recover the message signal contained in the received signal. If the message signal is transmitted by carrier modulation, the receiver performs carrier demodulation to extract the message from the sinusoidal carrier.

Channel

The communication channel (transmission medium) is the physical medium that is utilized to send the signal from the transmitter to the receiver.

In wireless transmission, such as microwave radio, the transmission medium is usually the atmosphere or free space. Telephone channels, on the other hand, employ a variety of physical media such as wire lines and optical fiber cables.

Irrespective of the type of physical medium for signal transmission, the essential feature is that the transmitted signal is corrupted in a random manner by a variety of possible mechanisms.

 The effects of these phenomena (attenuation, distortion, interference, noise, etc.) are shown at the center of Figure, since the transmission medium is often the most vulnerable part of a communication system, particularly over long distances.

Attenuation is  caused by losses within the system, it reduces the size or strength of the signal.

Whereas distortion is any alteration of the waveshape itself due to energy storage and/or non linearity.

Interference is caused by contamination by extraneous signals.

Noise is generated from sources that are internal and external to the system.

Common Signal Processing Operations

Amplification to compensate for attenuation

Filtering to reduce interference and noise, and/or to obtain selected facets of information

Equalization to correct some types of distortion

Frequency translation or sampling to get a signal that better suits the system characteristics

 Multiplexing to permit one transmission system to handle two or more information-bearing signals simultaneously.

COMPARISON OF ANALOG AND DIGITAL COMMUNICATION

Analog signals in an analog communication system can be transmitted directly via carrier modulation over the communication channel and demodulated accordingly at the receiver.

Alternatively, an analog source output may be converted into a digital form and the message can be transmitted via digital modulation and demodulated as a digital signal at the receiver.

Potential advantages in transmitting an analog signal by means of digital modulation are the following:  Signal fidelity is better controlled through digital transmission than

through analog transmission; effects of noise can be reduced significantly.

Since the analog message signal may be highly redundant, with digital processing, redundancy may be removed prior to modulation.

Digital communication systems are often more economical to implement.

DIGITAL COMMUNICATION SYSTEM

Figure illustrates the basic elements of a digital communication system. For eachfunction in the transmitting station, there is an inverse operation in the receiver.

The analog input signal (such as an audio or video signal) must first be converted to a digital signal by an analog to-digital (A/D) converter.

If no analog message is involved, a digital signal (such as the output of a teletype machine, which is discrete in time and has a finite number of output characters) can be directly input.

Encoding is a critical function in all digital systems. The messages produced by the source are usually converted into a sequence of binary digits. The process of efficiently converting the output of either an analog or a digital source into a sequence of binary digits is called source encoding or data compression.

The sequence of binary digits from the source encoder, known as the information sequence, is passed on to the channel encoder. The purpose of the channel encoder is to introduce some redundancy in a controlled manner in the binary information sequence, so that the redundancy can be used at the receiver to overcome the effects of noise and interference encountered in the transmission of the signal through the channel. Thus, redundancy in the information sequence helps the receiver in decoding the desired information sequence, thereby increasing the reliability of the received data and improving the fidelity of the received signal.

The binary sequence at the output of the channel encoder is passed on to the digital modulator, which functions as the interface to the communication channel. The primary purpose of the digital modulator is to map the binary information sequence into signal waveforms, since nearly all the communication channels used in practice are capable of transmitting electric signals (waveforms). Because the message has only two amplitudes in a binary system, the modulation process is known as keying.

In amplitude-shift keying (ASK), a carrier’s amplitude is shifted or keyed between two levels. Phase-shift keying (PSK) involves keying between two phase angles of the carrier, whereas frequency-shift keying (FSK) consists of shifting a carrier’s frequency between two values.

At the receiving end of a digital communication system, the digital demodulator processes the channel-corrupted transmitted waveform and reduces each waveform to a single number, which represents an estimate of the transmitted data symbol.

For example, when binary modulation is  used, the demodulator may process the received waveform and decide on whether the transmitted bit is a 0 or 1.

The source decoder accepts the output sequence from the channel decoder, and from the knowledge of the source encoding method used, attempts to reconstruct the original signal from the source.

Errors due to noise, interference, and practical system imperfections do occur.

The digital-to-analog (D/A) converter reconstructs an analog message that is a close approximation to the original message.

The difference between the original signal and the reconstructed signal is a measure of the distortion introduced by the digital communication system.

PERIODIC AND NONPERIODIC SIGNAL

A periodic signal has the property that it repeats itself in time, and hence, it is sufficient to

specify the signal in the basic time interval called the period.

A periodic signal x(t) satisfies the propertyx(t + kT ) = x(t)

for all t, all integers k, and some positive real number T, called the period of the signal. For discrete-time periodic signals, it follows that

x(n + kN) = x(n) for all integers n, all integers k, and a positive integer N,

called the period. A signal that does notsatisfy the condition of periodicity is known as non-periodic.

Examples

Periodic Signal Nonperiodic Signal

Discrete Time Periodic Signal

This is not periodic for all values of f. The condition for it to be periodic is2πf(n + kN) + θ = 2πfn + θ + 2mπ

for all integers n and k, some positive integer N, and some integer m. From this it follows that

2πfkN = 2mπ or f = m/(kN)that is, the discrete sinusoidal signal is periodic onlyfor rational values of f.

Even and Odd Signals

Evenness and oddness are expressions of various types of symmetry present in signals.

A signal x(t) is even if it has a mirror symmetry with respect to the vertical axis. The signal x(t) is even if and only if, for all t, it satisfies

x(−t) = x(t) A signal is odd if it is symmetric with respect to the origin  and only if,

for all t,x(−t) = −x(t)

Cosine wave is even signal while sine wave is odd signal as shown in figure below.

Any signal x(t), in general, can be expressed as the sum of its even and odd parts,

x(t) = xe(t) + xo(t)xe(t) = (x(t) + x(−t))/2xo(t) = (x(t) − x(−t))/2

The half-wave symmetry is expressed byx(t+- T/2) = -x(t)

Causal and Noncausal Signals

A signal x(t) is said to be causal if, for all t < 0, x(t) = 0; otherwise, the signal is noncausal.

An anticausal signal is identically equal to zero for t > 0.

A discrete-time signal is a causal signal if it is identically equal to zero for n < 0.

Note that the unit step multiplied by any signal produces a causal version of the signal.

Energy and Power Signals

Signals can also be classified as energy-type and power-type signals based on the finiteness of their energy content and power content, respectively.

A signal x(t) is an energy-type signal if and only if the energy Ex of the signal,

Hence, the given x(t) is a power-type signal.

Periodic signals are not typically energy type. The power content of a periodic signal is equal to the average power in one period. So they are power signals.