Digital signal Processing By Dileep Kumar dk_2kes21@yahoo.com

Lecture 1-2 Basic Concepts

Signal: A signal is defined as a function of one or more variables which conveys information on the nature of a physical phenomenon. The value of the function can be a real valued scalar quantity, a complex valued quantity, or perhaps a vector. System: A system is defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.

Continuos-Time Signal: A signal x(t) is said to be a continuous time signal if it is defined for all time t. Discrete-Time Signal: A discrete time signal x[nT] has values specified only at discrete points in time. Signal Processing: A system characterized by the type of operation that it performs on the signal. For example, if the operation is linear, the system is called linear. If the operation is non-linear, the system is said to be non-linear, and so forth. Such operations are usually referred to as “Signal Processing”.

Basic Elements of a Signal Processing System Analog output signal Analog input signal Analog Signal Processor Analog Signal Processing Analog input signal Analog output signal A/D converter Digital Signal Processor D/A converter Digital Signal Processing

Advantages of Digital over Analogue Signal Processing: A digital programmable system allows flexibility in reconfiguring the DSP operations simply by changing the program. Reconfiguration of an analogue system usually implies a redesign of hardware, testing and verification that it operates properly. DSP provides better control of accuracy requirements. Digital signals are easily stored on magnetic media (tape or disk). The DSP allows for the implementation of more sophisticated signal processing algorithms. In some cases a digital implementation of the signal processing system is cheaper than its analogue counterpart.

DSP Applications Space Medical Telephone Space photograph enhancement Data compression Intelligent sensory analysis Space Medical image storage and retrieval Medical Image and sound compression for multimedia presentation. Movie special effects Video conference calling Commercial Video and data compression echo reduction signal multiplexing filtering Telephone

Classification of Signals Deterministic Signals A deterministic signal behaves in a fixed known way with respect to time. Thus, it can be modeled by a known function of time t for continuous time signals, or a known function of a sampler number n, and sampling spacing T for discrete time signals. Random or Stochastic Signals: In many practical situations, there are signals that either cannot be described to any reasonable degree of accuracy by explicit mathematical formulas, or such a description is too complicated to be of any practical use. The lack of such a relationship implies that such signals evolve in time in an unpredictable manner. We refer to these signals as random.

Even and Odd Signals A continuous time signal x(t) is said to an even signal if it satisfies the condition x(-t) = x(t) for all t The signal x(t) is said to be an odd signal if it satisfies the condition x(-t) = -x(t) In other words, even signals are symmetric about the vertical axis or time origin, whereas odd signals are antisymmetric about the time origin. Similar remarks apply to discrete-time signals. Example: even odd odd

Periodic Signals A continuous signal x(t) is periodic if and only if there exists a T > 0 such that x(t + T) = x(t) where T is the period of the signal in units of time. f = 1/T is the frequency of the signal in Hz. W = 2/T is the angular frequency in radians per second. The discrete time signal x[nT] is periodic if and only if there exists an N > 0 such that x[nT + N] = x[nT] where N is the period of the signal in number of sample spacings. Example: Frequency = 5 Hz or 10 rad/s 0.2 0.4

Continuous Time Sinusoidal Signals A simple harmonic oscillation is mathematically described as x(t) = Acos(wt + ) This signal is completely characterized by three parameters: A = amplitude, w = 2f = frequency in rad/s, and  = phase in radians. A T=1/f

Discrete Time Sinusoidal Signals A discrete time sinusoidal signal may be expressed as x[n] = Acos(wn + ) - < n <  Properties: A discrete time sinusoid is periodic only if its frequency is a rational number. Discrete time sinusoids whose frequencies are separated by an integer multiple of 2 are identical. The highest rate of oscillation in a discrete time sinusoid is attained when w =  ( or w = - ), or equivalently f = 1/2 (or f = -1/2). 2 4 6 8 10 -1 1

Energy and Power Signals A signal is referred to as an energy signal, if and only if the total energy of the signal satisfies the condition 0 < E <  On the other hand, it is referred to as a power signal, if and only if the average power of the signal satisfies the condition 0 < P <  An energy signal has zero average power, whereas a power signal has infinite energy. Periodic signals and random signals are usually viewed as power signals, whereas signals that are both deterministic and non-periodic are energy signals.

Basic Operations on Signals (a) Operations performed on dependent variables 1. Amplitude Scaling: let x(t) denote a continuous time signal. The signal y(t) resulting from amplitude scaling applied to x(t) is defined by y(t) = cx(t) where c is the scale factor. In a similar manner to the above equation, for discrete time signals we write y[nT] = cx[nT] 2x(t) x(t)

Example: AM Radio Signal 2. Addition: Let x1 [n] and x2[n] denote a pair of discrete time signals. The signal y[n] obtained by the addition of x1[n] + x2[n] is defined as y[n] = x1[n] + x2[n] Example: audio mixer 3. Multiplication: Let x1[n] and x2[n] denote a pair of discrete-time signals. The signal y[n] resulting from the multiplication of the x1[n] and x2[n] is defined by y[n] = x1[n].x2[n] Example: AM Radio Signal

(b) Operations performed on independent variable Time Scaling: Let y(t) is a compressed version of x(t). The signal y(t) obtained by scaling the independent variable, time t, by a factor k is defined by y(t) = x(kt) if k > 1, the signal y(t) is a compressed version of x(t). If, on the other hand, 0 < k < 1, the signal y(t) is an expanded (stretched) version of x(t).

Example of time scaling 1 Expansion and compression of the signal e-t. 0.9 0.8 exp(-t) 0.7 0.6 0.5 exp(-2t) 0.4 exp(-0.5t) 0.3 0.2 0.1 5 10 15

Time scaling of discrete time systems 10 x[n] 5 -3 -2 -1 1 2 3 10 x[0.5n] 5 -1.5 -1 -0.5 0.5 1 1.5 5 x[2n] -6 -4 -2 2 4 6 n

Time Reversal This operation reflects the signal about t = 0 and thus reverses the signal on the time scale. 5 x[n] 1 2 3 4 5 n x[-n] -5 1 2 3 4 5 n

Time Shift A signal may be shifted in time by replacing the independent variable n by n-k, where k is an integer. If k is a positive integer, the time shift results in a delay of the signal by k units of time. If k is a negative integer, the time shift results in an advance of the signal by |k| units in time. 1 x[n] 0.5 -2 2 4 6 8 10 1 x[n+3] 0.5 -2 2 4 6 8 10 1 x[n-3] 0.5 -2 2 4 6 8 10 n