Signals And Systems- Unit -i

SIGNALS AND SYSTEMS UNIT –I

TYPES OF SIGNALS • Two types of signals • (i) Analog signal The input , out put and the system will be also analog in nature • (ii) Digital signal The input ,out put and the system will be also digital in nature Basic elements of digital signal processing i/p signal O/PSIGNAL A/D converter

DIGITAL SIGNAL PROCESSOR

D/A CONVTER

• A/D CONVERTER: converts analog signal to digital signal .The A/D converts determine sampling rate quantization error • DSP PROCESSOR: It consist of ALU ,shifter ,interrupts etc . Its like microprocessor but it is implemented fast • D/A CONVERTER: some of the processed signals are required back in their analog form so we are using D/A converter comparison between DSP and analog signal processing (i) flexibility (ii) accuracy (iii) easy storage (iv) mathematical processing (v) cost (iv) size and realibility

• DSP APPLICATION (i) DSP for voice and speech : speech recognition ,voice mail (ii) DSP for telecommunication : FAX , cellular phone speaker (iii) DSP for consumer application : digital audio /video /television (iv) DSP for graphics and image : 3D and 2D visualization animation

(v) DSP for military : Rader processing sonar processing etc (vi) DSP for biomedical : x-ray, CT scan, ECG (vii) DSP for industrial : Robotics ,security access

CLASIFICATION OF SIGNALS • 1. 2. 3. 4. • 1. 2. 3. 4.

The signals are as Analog and digital signals Continuous and discrete amplitude signals Continuous and discrete time signals Multi channel and multi dimensional signals Signals are further classified on the basics of properties Periodic and non periodic signals Energy and power signals Even and odd signals Deterministic and random signals

Continuous and discrete time signals The signal is are defined for every instant of time are know as continuous –time signal . They are denoted by x (t) The signal are defined for every discrete instants of time . The discrete time signal are continuous in amplitude and discrete in time and it is denoted by x (n) Continuous time signals Discrete time signals x (t)=x (n T) =e -nT X (t)=e –t

-T 0

0

t

T

2T

3T

Periodic and non periodic signals The signal x (n) is said to be periodic if it repeats after certain period x (n + N) =x (n) for all values of n Here N is period of x (n ) samples . Now consider a cosine wave signal x (n) = A cos (2 Π f o n) Here x (n + N) in above equ x (n + N) =A cos (2 Π f o N + n) =A cos (2 Π f o N + 2 Π f on ) A cos (2 Π f o N + 2 Π f on ) = A cos (2 Π f o n) The above equ is satisfied only if 2 Π f o is an integer multiple of 2 Π Here k is some integer solving for f o the above equ becomes f o=k/N the signal is only periocic only if f o is rational

Energy and power signals The energy signal of the discrete time signal x (n) is denoted by E its given by ∞ 2 energy E =

∑Ιx(n)Ι

If x (n) is finite n=−∞ X (n) is energy signal if ,0< E <∞ The average power of the discrete tome signal x (n) is denoted by P It is given as ∞ 1 Power ,p = lim (2N + 1) Ιx(n)Ι2 ∞

∑

n=−∞

There are few signal which r neither energy are power signal

DISCRETE TIME SYSTEM SYSTEM : the discrete system is a device that performs some operation on the discrete signal the out put response for the discrete time signal is represented by y (n) = T [x (n)] or x (n) T y (n) T represents transformation REPRESENTATION OF DISCRETE TIME SYSTEM ADDER: CONSTANT MULTIPLER: X1(n) X (n) +

X2(n)

y (n) = X1(n) + X2(n)

a

y (n)= X (n)

signal multiplier :

X1(n)

y= X1(n) . X2(n)

x

X2(n)

Unit advance block

x (n)

Z

y (n) = X (n+1)

unit block delay :

x (n)

y (n) = X (n-1) Z-1

CLASSIFICATION OF DISCRETE TIME SYSTEMS 1. Static and dynamic systems 2. Time invariant and time variant systems 3. Linear and non linear systems 4. Casual and non casual system 5. Stable and un stable systems Static and dynamic systems: when the output of the system depends only on the present input samples then it is static system eg : y (n) = x (n) When the out of the system depend on the pass out put of the system then its dynamic Eg : y (n) = x (n) + x (n-1)

•

Time invariant and time variant systems : The input and out put of the system will not change in shift in time eg : x (n) T y (n) x (n- k) T y (n- k) Linear and non linear systems : a system is said to be linear if it satisfies super position principle . T{a1 X1(n) +a2 X2(n) } = a1T[X1(n)] +a2T [X2(n) ]

X1(n)

a1

y (n) =T{a1 X1(n) +a2 X2(n) } + X2(n) a2

Discrete time system

•

X1(n) Discrete

a1

system + X 2(n)

Discrete system

y’ (n) =a1T[X1(n)] +a2T [X2(n) ]

a2

Casual and non casual system : in the casual system the out put depend upon the past and present inputs only X (n), x (n-1) The system is non casual if its out put depends upon the future inputs also x (n+1)