Module 2: Signals and Systems PDF

Depending on the nature of sampling, quantization, and other characteristics, signals are classified. Classification helps to choose the most appropriate processing for a given signal. Unnecessary work and possible errors can be avoided, if the signal characteristics are known. A signal is defined...

Depending on the nature of sampling, quantization, and other characteristics, signals are classified. Classification helps to choose the most appropriate processing for a given signal. Unnecessary work and possible errors can be avoided, if the signal characteristics are known. A signal is defined as any physical quantity that varies with time, space, or any other independent variable or variables. Mathematically, we describe a signal as a function of one or more independent variables. For example, the functions s~1~(t) = 5t s~2~(t) = 20t^2^ describe two signals, one that varies linearly with the independent variable t (time) and a second that varies quadratically with t. As another example, consider the function s (x,y) = 3x + 2xy + 10y^2^ This function describes a signal of two independent variables x and y that could represent the two spatial coordinates in a plane. The signals described belong to a class of signals that are precisely defined by specifying the functional dependence on the independent variable. However, there are cases w here such a functional relationship is unknown or too highly complicated to be of any practical use.   Continuous-Time Signals and Discrete-Time Signal This type, also called analog signals, is characterized by its continuous nature of the dependent and independent variables. For example, the room temperature is a continuous signal. It has a value at any instant of time. The mathematical form of a signal is x(t), where t is the independent variable and x(t) is the dependent variable. That is, x(t) is a function of t. While the independent variable for most of the signals is time, it could be any other entity such as frequency or distance. The mathematical characterization is the same irrespective of the nature of the signal. It could be a temperature signal or a pressure signal or anything else. Figure 1.1a shows an arbitrary continuous-time signal. This is a typical example of signals occurring in practice with arbitrary amplitude profile. In digital signal processing, the first step, as mentioned earlier and presented later, is to approximate such signals in terms of basic signals so that the processing is simplified. The signal is defined at each and every instant of time over the period of its occurrence. It can assume any real or complex value at each instant. While all practical signals are real-valued, complex-valued signals are necessary and often used as intermediaries in signal processing. Fig. 2.1 (a) Continuous-time signal; (b) discrete-time signal; (c) quantized continuous-time signal; (d) digital signal In this type of signals, also called discrete signals, the values of the signal are available only at discrete intervals. That is, we take the sample values of a continuous signal at intervals of Ts, the sampling interval. The signal could also be inherently a discrete-time signal. Irrespective of the source, the discrete value x(n) is called the nth sample of the signal. The sampling interval Ts is usually suppressed. Figure 1.1b shows the discrete-time signal corresponding to the signal shown in Fig. 1.1a. The mathematical form of this type of signal is x(nTs), where nTs is the independent variable and x(nTs) is the dependent variable. That is, x(nTs) is a function of nTs. In the expression nTs, the sampling interval Ts is assumed to be a constant in this book and the range of the integer variable, in general, is n = −∞,..., −1, 0, 1,..., ∞. It looks like that, we cannot recover x(t) from x(nTs) as lots of the values of the signal between sampling intervals are lost in the sampling process. It is true, in this case, as the sampling interval is not short enough to represent and process it with its samples. However, any practical signal, with sufficiently short sampling interval, can be represented by a finite number of samples and reconstructed after processing with adequate accuracy. Practical signals can be considered, with negligible error, as composed of frequency components with frequencies varying over a finite range (band-limited). The sampling theorem says that any continuous signal can be exactly reconstructed from its sampled version with a sampling interval T\ s, such that more than two samples in a cycle of the highest frequency component are taken. The sampling theorem is the basis on which DSP is based. The reason we want to process the signal using its samples, rather than the continuous version itself, is that DSP includes the advantages of low-cost, higher noise immunity, and highly reliable digital components. The disadvantages are the errors accumulating due to the representation of numbers by a finite number of bits and the limitation of the frequency of operation due to sampling. In most applications, the advantages outweigh the disadvantages. Therefore, DSP is widely used in applications of science and engineering.   Periodic and Aperiodic Signals A signal which has a definite pattern and repeats itself at regular intervals of time is called a periodic signal, and a signal which does not repeat at regular intervals of time is called a non-periodic or aperiodic signal. A discrete-time signal x(n) is said to be periodic if it satisfies the condition x(n) = x(n + N) for all integers n. The smallest value of N which satisfies the above condition is known as a fundamental period. If the above condition is not satisfied even for one value of n, then the discrete-time signal is aperiodic. Sometimes aperiodic signals are said to have a period equal to infinity. The angular frequency is given by 𝜔 = 2𝜋/𝑁 Fundamental period N = 2𝜋 /𝜔 The sum of two discrete-time periodic sequences is always periodic.   Even and Odd Signals A discrete-time signal x(n) is said to be an even (symmetric) signal if it satisfies the condition: x(n) = x(--n) for all n Even signals are symmetrical about the vertical axis or time origin. Hence they are also called symmetric signals: cosine sequence is an example of an even signal. Some even signals are shown in Figure. An even signal is identical to its reflection about the origin. For an even signal x0(n) = 0. A discrete-time signal x(n) is said to be an odd (anti-symmetric) signal if it satisfies the condition: x(--n) = --x(n) for all n Odd signals are anti-symmetrical about the vertical axis. Hence they are called anti- symmetric signals. Sinusoidal sequence is an example of an odd signal. For an odd signal xe(n) = 0. Some odd signals are shown in Figure. Fig 2.2 Example of even (a) and odd (b) signals   Energy and Power Signals Signals may also be classified as energy signals and power signals. However there are some signals which can neither be classified as energy signals nor power signals. The total energy E of a discrete-time signal x(n) is defined as:    While the average power P of a discrete-time signal x(n) is defined as: A signal is said to be an energy signal if and only if its total energy E over the interval (-- ∞, ∞) is finite (i.e., 0 \< E \< ∞). For an energy signal, average power P = 0. Non-periodic signals which are defined over a finite time (also called time limited signals) are the examples of energy signals. Since the energy of a periodic signal is always either zero or infinite, any periodic signal cannot be an energy signal. A signal is said to be a power signal, if its average power P is finite (i.e., 0 \< P \< ∞). For a power signal, total energy E = ∞. Periodic signals are the examples of power signals. Every bounded and periodic signal is a power signal. But it is true that a power signal is not necessarily a bounded and periodic signal. Both energy and power signals are mutually exclusive, i.e. no signal can be both energy signal and power signal. The signals that do not satisfy the above properties are neither energy signals nor power signals. For example, x(n) = u(n), x(n) = nu(n), x(n) = n 2 u(n). These are signals for which neither P nor E are finite. If the signals contain infinite energy and zero power or infinite energy and infinite power, they are neither energy nor power signals. If the signal amplitude becomes zero as \|n\| → ∞, it is an energy signal, and if the signal amplitude does not become zero as \|n\| → ∞, it is a power signal.   Causal and Noncausal Signals A discrete-time signal x(n) is said to be causal if x(n) = 0 for n \< 0, otherwise the signal is noncausal. A discrete-time signal x(n) is said to be anti-causal if x(n) = 0 for n \> 0. A causal signal does not exist for negative time and an anti-causal signal does not exist for positive time. A signal which exists in positive as well as negative time is called a non-casual signal. The u(n) is a causal signal and u(-- n) an anti-causal signal, whereas x(n) = 1 for -- 2 ≤ n ≤ 3 is a non-causal signal.   Deterministic and Random Signals A signal exhibiting no uncertainty of its magnitude and phase at any given instant of time is called deterministic signal. A deterministic signal can be completely represented by mathematical equation at any time and its nature and amplitude at any time can be predicted. A signal characterized by uncertainty about its occurrence is called a non-deterministic or random signal. A random signal cannot be represented by any mathematical equation. The behavior of such a signal is probabilistic in nature and can be analyzed only stochastically. The pattern of such a signal is quite irregular. Its amplitude and phase at any time instant cannot be predicted in advance. A typical example of a non-deterministic signal is thermal noise.   Bounded Signal An arbitrary relaxed system is said to be bounded input-bounded output (BIBO) stable if and only if every bounded input produces a bounded output. There are several elementary signals which play vital role in the study of signals and systems. These elementary signals serve as basic building blocks for the construction of more complex signals. In fact, these elementary signals may be used to model a large number of physical signals, which occur in nature. These elementary signals are also called standard signals. The standard discrete-time signals are as follows: 1\. Unit step sequence 2\. Unit ramp sequence 3\. Unit impulse sequence 4\. Sinusoidal sequence 5\. Exponential sequence   Unit Step Sequence The step sequence is an important signal used for analysis of many discrete-time systems. It exists only for positive time and is zero for negative time. It is equivalent to applying a signal whose amplitude suddenly changes and remains constant at the sampling instants forever after application. In between the discrete instants it is zero. If a step function has unity magnitude, then it is called unit step function. The usefulness of the unit-step function lies in the fact that if we want a sequence to start at n = 0, so that it may have a value of zero for n \< 0, we only need to multiply the given sequence with unit step function u (n). The discrete-time unit step sequence u (n) is defined as: The shifted version of the discrete-time unit step sequence u(n -- k) is defined as: It is zero if the argument (n -- k) \< 0 and equal to 1 if the argument (n -- k) S 0. The graphical representation of u (n) and u (n -- k) is shown in Figure 2.3 Fig 2.3 Discrete--time (a) Unit step function (b) Shifted unit step function    Unit Ramp Sequence The discrete-time unit ramp sequence r (n) is that sequence which starts at n = 0 and increases linearly with time and is defined as: or It starts at n = 0 and increases linearly with n. The shifted version of the discrete-time unit ramp sequence r(n -- k) is defined as: The graphical representation of r(n) and r(n -- 2) is shown in Figure 2.4 Fig 2.4 Discrete--time (a) Unit ramp sequence (b) Shifted ramp sequence.   Unit Impulse Function or Unit Sample Sequence The discrete-time unit impulse function (n), also called unit sample sequence, is defined as: This means that the unit sample sequence is a signal that is zero everywhere, except at n = 0, where its value is unity. It is the most widely used elementary signal used for the analysis of signals and systems. The shifted unit impulse function (n -- k) is defined as:  The graphical representation of (n) and (n -- k) is shown in Figure 2.5 Fig 2.5 Discrete--time (a) Unit sample sequence (b) Delayed unit sample sequence   Properties of discrete-time unit sample sequence The unit sample sequence 𝛿(n) and the unit step sequence u(n) are related as:   Sinusoidal Sequence The discrete-time sinusoidal sequence is given by X(n) = A sin (𝜔𝑛 + ∅) where A is the amplitude, 𝜔 is angular frequency, ∅ is phase angle in radians and n is an integer. The period of the discrete-time sinusoidal sequence is: N = \[2𝜋/𝜔\] 𝑚 where N and m are integers. All continuous-time sinusoidal signals are periodic, but discrete-time sinusoidal sequences may or may not be periodic depending on the value of. For a discrete-time signal to be periodic, the angular frequency must be a rational multiple of 2. The graphical representation of a discrete-time sinusoidal signal is shown in Figure 2.6 Fig 2.6 Discrete-time sinusoidal signal   Exponential Sequence The discrete-time real exponential sequence a^n^ is defined as: X(n) = a^n^ for all n Figure 2.7 illustrates different types of discrete-time exponential signals. - When 0 \< a \< 1, the sequence decays exponentially as shown in Figure 2.7 (a). - When a \> 1, the sequence grows exponentially as shown in Figure 2.7 (b). - When a \< 0, the sequence takes alternating signs as shown in Figure 2.7 (c) and (d). Fig 2.7 Graphical representation of exponential signals. But when the parameter a is complex valued, it can be expressed as where r and ∅ are now the parameters. Hence we can express x(n) as  When we process a sequence, this sequence may undergo several manipulations involving the independent variable or the amplitude of the signal. The basic operations on sequences are as follows: 1\. Time shifting 2\. Time reversal 3\. Time scaling 4\. Amplitude scaling 5\. Signal addition 6\. Signal multiplication The first three operations correspond to transformation in independent variable n of a signal. The last three operations correspond to transformation on amplitude of a signal.   Time Shifting The time shifting of a signal may result in time delay or time advance. The time shifting operation of a discrete-time signal x(n) can be represented by y(n) = x(n -- k) This shows that the signal y (n) can be obtained by time shifting the signal x(n) by k units. If k is positive, it is delay and the shift is to the right, and if k is negative, it is advance and the shift is to the left. An arbitrary signal x(n) is shown in Figure 2.8 (a). x(n -- 3) which is obtained by shifting x(n) to the right by 3 units (i.e. delay x(n) by 3 units) is shown in Figure 2.8 (b). x(n + 2) which is obtained by shifting x(n) to the left by 2 units (i.e. advancing x(n) by 2 units) is shown in Figure 2.8 (c). Fig 2.8 (a) Sequence x(n) (b) x(n -- 3) (c) x(n + 2).   Time Reversal The time reversal also called time folding of a discrete-time signal x(n) can be obtained by foldingthe sequence about n = 0. The time reversed signal is the reflection of the original signal. It is obtained by replacing the independent variable n by --n. Figure 2.9(a) shows an arbitrary discrete-time signal x(n), and its time reversed version x(--n) is shown in Figure 2.9(b). Figure 2.9\[(c) and (d)\] shows the delayed and advanced versions of reversed signal x(--n). The signal x(--n + 3) is obtained by delaying (shifting to the right) the time reversed signal x(--n) by 3 units of time. The signal x(--n -- 3) is obtained by advancing (shifting to the left) the time reversed signal x(--n) by 3 units of time. Fig 2.9 shows other examples for time reversal of signals   Time Scaling Time scaling may be time expansion or time compression. The time scaling of a discrete- time signal x(n) can be accomplished by replacing n by an in it. Mathematically, it can be expressed as: y(n) = x(an) when a \> 1, it is time compression and when a \< 1, it is time expansion Let x(n) be a sequence as shown in Figure 2.10(a). If a = 2, y(n) = x(2n). Then y(0) = x(0) = 1 y(--1) = x(--2) = 3 y(--2) = x(--4) = 0 y(1) = x(2) = 3 y(2) = x(4) = 0 and so on. So to plot x(2n) we have to skip odd numbered samples in x(n). Fig 2.10 Discrete--time scaling (a ) Plot of x(n) (b) Plot of x(2n )    Amplitude Scaling The amplitude scaling of a discrete-time signal can be represented by y(n) = ax(n) where a is a constant. The amplitude of y(n) at any instant is equal to a times the amplitude of x(n) at that instant. If a \> 1, it is amplification and if a \< 1, it is attenuation. Hence the amplitude is rescaled. Hence the name amplitude scaling. Figure 2.11 (a) shows a signal x(n) and Figure 2.11(b) shows a scaled signal y(n) = 2x(n).   Signal Addition In discrete-time domain, the sum of two signals x1(n) and x2(n) can be obtained by adding the corresponding sample values and the subtraction of x2(n) from x1(n) can be obtained by subtracting each sample of x2(n) from the corresponding sample of x1(n) as illustrated below. If x1(n) = {1, 2, 3, 1, 5} and x2(n) = {2, 3, 4, 1, --2} Then x1(n) + x2(n) = {1 + 2, 2 + 3, 3 + 4, 1 + 1, 5 -- 2} = {3, 5, 7, 2, 3} and x1(n) -- x2(n) = {1 -- 2, 2 -- 3, 3 -- 4, 1 -- 1, 5 + 2} = {--1, --1, --1, 0, 7} 1.4.6   Signal multiplication The multiplication of two discrete-time sequences can be performed by multiplying their values at the sampling instants as shown below. If x1(n) = {1, --3, 2, 4, 1.5} and x2(n) = {2, --1, 3, 1.5, 2} Then x1 (n) x2 (n) = {1 × 2,- 3 ×-1, 2 × 3, 4 × 1.5, 1.5 × 2} = {2, 3, 6, 6, 3}