Analog-To-Digital convertorSamplerQuantizationCoding.

 

Sampler Quantizes Codes

Discrete time signal Digital signal

Quantized signal Analog signal

Sampling of analog signal This is the conversion of a continuous-time signal

into discrete-time signal obtained by taking samples of the continuous-time signal at discrete-

time instant

The sampling theorem introduced by Nyquist and later by Shannon is used to determine the appropriate sampling rateFs≥2Fmax

Quantization of Continues –Amplitude Signals

)()()( nxnxne qq

Quantizer convert the continouse amplitude into finite sets of discrete value called quantization level There is an error due to this conversation called quantization error

2)(

2

neq

1minmax

L

xx

…

The distance between the two levels is Δ and its computed as

Example (1-3):- Consider the analog signal x(t) is an input to analog to digital convertor ,what is the output of sampler and quantizer if the sampling rate is 1Hz. Let the number of quantization level =11Where , take only 9 samples.

Coding of Quantized SamplesWith a word length of b bit we can create different binary numbers

Digital-To-Analog converters

Discrete time signals

representation of the discrete time signal 1. Functional representation, such as

otherwise

nfornfor

nx0

243,11

)(

2. Tabular representation , such as

)(nx n … -2 -1 0 1 2 3 …..

… 0 0 0 1 4 1 …. …..

3. Sequence representation An infinite-duration signal or sequence with the time origin (n=0) indicated by the symbol ↑ is represented as

  ,0,0,3,8,1,4,1,0,0,)( nx

Some elementary discrete time signals

1. The unit sample sequence is denoted as δ(n) and is defined as

0001

)(nn

n

2. Unit step signal- it is denoted as u(n) and is defined as

0001

)(nn

nu

3.Unit ramp signal :-it is denoted as r(n)and is defined as

000

)(n

nnnr … (2.9)

4.Exponential function: - an exponential sequence is defined by nanx )( … (2.10)

Signal Manipulations a. Transformation of the independent variable (time) Shifting Reversal Time Scaling

b. Amplitude modification of discrete time signalAmplitude scalingThe sum of two signalsThe product of two signals

H.w : For the signal shown below ,sketch the following signalsX(n-2)X(n+3).δ(n-2)X(n-1)+u(-n+1)

Classification of discrete time signal Even and Odd signals:-

A real-valued signal is said to be even if for all nx(n) = x(-n) Whereas a signal is said to be odd if for all nx(n) = -x(-n)

X(n)=xe(n)+xo(n) To find the even part of x(n)

Whereas to find the odd part of x(n)

Example 2.3: - Find the even and odd parts of the following signal:

Periodic and Aperiodic Sequences

A signal x(n) is said to be periodic if and only if x(n)=x(n+N) for all n.

Example 2.4: Determine if the following signal is periodic or not then find its period