Download presentation

Presentation is loading. Please wait.

Published byShannon Lonon Modified over 2 years ago

1
**Wireless Access Systems: Introduction and Course Outline**

2
**Some Wireless Access Systems**

Wireless Access Systems provide short to medium range tetherless access to a backhaul network, a central unit or peer nodes Examples include Bluetooth WLAN Vehicular Networks WiMax RFID WBAN WPAN Communication Technology Laboratory Wireless Communication Group

3
**Wireless Access Systems are Ubiquitous**

Internet Communication Technology Laboratory Wireless Communication Group

4
**Some More Applications**

intelligent home ambient intelligence security wearable computing shopping defence surveillance traffic security surveillance access control supply chain management environment Internet logistics industrial communications instant messaging enterprise communication information exchange pervasive computing virtual reality health care home care Communication Technology Laboratory Wireless Communication Group

5
**Characteristics of Wireless Access Systems**

Heterogeneous standards IEEE WLAN IEEE WPAN IEEE WMAN (Hiperlan) Bluetooth DECT various RFID standards RFID tags, readers sensors, actors communication appliances information access information processing backhaul access points Heterogeneous nodes Lots of spectrum (approx.) (ISM) (ISM) (ISM) (ISM) (UWB) WLAN Internet backhaul Sensor network RFID cellular: GSM UMTS WPAN WMAN Pervasive wireless access Communication Technology Laboratory Wireless Communication Group

6
**The Throughput - Range Tradeoff**

RFID Body Area Networks 100M 10M 1M 100k 10k 1k 1 3 10 30 100 range [m] link throughput [bps] 11b 11a 11g 15.4 15.3 15.1 Sensor Networks 15.3a WLAN WPAN UWB Bluetooth ZigBee UWB region (conceptional) Communication Technology Laboratory Wireless Communication Group

7
**Outline of Course Fundamentals**

Fundamentals of short/medium range wireless communication 1 digital transmission systems equivalent baseband model digital modulation and ML-detection Fundamentals of short/medium range wireless communication 2 fading channels diversity MIMO wireless Fundamentals of short/medium range wireless communication 3 Multicarrier modulation and OFDM Systems I: OFDM based broadband access WLAN 1: IEEE g, a WLAN 2: IEEE n WMAN: (mobile) WiMAX Vehicular Networks Systems II: Wireless short range access technolgies and systems UWB 1: Promises and challenges of Ultra Wideband Systems UWB 2: Physical Layer options Wireless Body Area Network case study: UWB based human motion tracking The IEEE x family of Wireless Personal Area Networks (WPAN): Bluetooth, ZigBee, UWB Systems III: RF identification (RFID) and sensor networks RFID 1 RFID 2 Communication Technology Laboratory Wireless Communication Group

8
**Exercises: Motivation**

Simulate, practice, verify, learn and have fun We will simulate the theoretical ideas/methods/techniques that we learn throughout the lecture. MATLAB (matrix laboratory) will be used for simulations. In general we will simulate Single carrier transmission Multi-carrier transmission Wireless Channel Channel coding Simple UWB transceiver Communication Technology Laboratory Wireless Communication Group

9
**Exercises: Organization**

Students organize in 2 or 4 groups There will be three exercises with two tasks each during the semester. Each group will perform one of the two tasks and then present the results. The general schedule of tasks: Introduction of tasks. Working period (2 weeks). Present afterwards. Each group will work individually. Combining period (1 week). Present afterwards. Two groups will work in collaboration. For further details will be presented in the first exercise lecture next week 8:15 Communication Technology Laboratory Wireless Communication Group

10
**Schedule: Communication Technology Laboratory**

8:15-9:00 9:15-10:00 10:15-11:00 11:15-12:00 Week 1 Fundamentals of wireless communications. 1 Fundamentals of wireless communications. 1 Week 2 Introduction – First Exercise Fundamentals of wireless communications. 2 Week 3 Fundamentals of wireless communications. 3 Week 4 Presentation of Ex 1/ 1 Presentation of Ex 1/2 WLAN - 1 Week 5 optional: wrap up of simulation basics optional: revised solutions of Ex 1/1 and EX 1/2 WLAN - 2 Week 6 Introduction- Second Exercise Presentation of Ex 1 - Combination step WiMAX 1 Week 7 Vehicular Networks Week 8 Presentation of Ex 2/1 Presentation of Ex 2/2 UWB 1 Week 9 Third Exercise Presentation of Ex 2 - Combination step UWB 2 Week 10 WBAN Week 11 Presentation of Ex 3/1 Presentation of Ex 3/2 WPAN Week 12 Presentation of Ex 3 - Combination step RFID 1 Week 13 RFID 2 Communication Technology Laboratory Wireless Communication Group

11
**Wireless Access Systems: Fundamentals of Short Range Wireless Communications**

12
**Fundamentals of Short Range Wireless: Outline**

Digital transmission and detection on the AWGN channel digital transmission systems equivalent baseband model digital modulation and ML-detection Fading channels fading channels diversity MIMO wireless Modulation schemes for frequency selective channels multicarrier modulation Orthogonal Frequency Division Multiplexing (OFDM) Communication Technology Laboratory Wireless Communication Group

13
**Equivalent Baseband Representation**

Communication Technology Laboratory Wireless Communication Group

14
Narrowband Case: Equivalent Baseband Model with Bandpass Channel, Different TX and RX Lowpasses and Frequency/Phase Offset + Notation: Narrowband case: Notes f0 and are called the reference frequency and phase of the BB model for f0 = f1 the BB model is time-invariant (a filter) Communication Technology Laboratory Wireless Communication Group

15
**Narrowband Case: Relation of Physical Signals and Their Complex Baseband Representation**

Im{} Re{} Im{} Re{} Re{} Names and Notation: The spectrum of the analytic signal in terms of the physical signal is given by Communication Technology Laboratory Wireless Communication Group

16
**Transmission of Digital Information I: Generation of Finite Signal Sets (Modulation)**

Communication Technology Laboratory Wireless Communication Group

17
**General Block Diagram of a Digital Modulator**

The information bit vector contains N bit It is mapped onto a message index (i) We use a look-up table with 2N transmit waveforms The transmit signal is selected according to the message index The process of selecting a transmit signal according to an information bit vector is called modulation For finite N this structure models a block transmission Mapper Communication Technology Laboratory Wireless Communication Group

18
**Signal Space Representation of Digital Modulator**

The signal space is defined by a set of orthonormal basepulses The basepulses are stacked to form the basepulse vector orthonormality implies The signal space representation of the transmit signals is obtained by the projection we refer to as transmit symbol vector Look-up table Mapper Communication Technology Laboratory Wireless Communication Group

19
Linear Modulation For linear modulation schemes the transmit symbol vector is obtained by a linear transformation of the input symbol vector precoding matrix GTX Dramatically reduces the size of the look-up table general modulation: exponential growth with the number N of information symbols linear modulation: linear growth Communication Technology Laboratory Wireless Communication Group

20
**Some Popular Linear Modulation Schemes**

name symbol alphabet 2-PAM 4-QAM (QPSK) Communication Technology Laboratory Wireless Communication Group

21
**Filter Implementation of Linear Modulator: Nyquist Basepulses**

Nyquist basepulses (orthonormal) Nyquist criterion T t Communication Technology Laboratory Wireless Communication Group

22
**Transmission of Digital Information II: Transmission and Detection**

Communication Technology Laboratory Wireless Communication Group

23
**Additive White Gaussian Noise (AWGN) Channel**

For analytical tractability usually a white noise process is assumed for physical system models (real-valued signals) we have for complex baseband representations as used herein we have Channel is modelled as additive noise source In many cases of practical interest the noise can be characterized as zero mean stationary Gaussian random process w(t) any set of samples is jointy normally distributed autocorrelation function power density spectrum Communication Technology Laboratory Wireless Communication Group

24
**Sufficient Statistic and Symbol Discrete System Model**

Continuous time system model Bank of correlators generates the decision vector the decision vector is a sufficient statistics (for additive white Gaussian noise; AWGN) contains all information for the optimal estimation of the transmit symbol vector With the impulse correlation matrix we obtain the symbol discrete system model the elements of the noise vector are statistically independent and identically distributed Gaussian random variables Bank of correlators Symbol discrete system model Communication Technology Laboratory Wireless Communication Group

25
**Frequency Selective Channel**

The channel is represented by a filter h(t) and AWGN A channel matched filter is required prior to the correlator bank in order to obtain a sufficient statistics These filters may affect the resulting impulse correlation matrix intersymbol interference (ISI) channel channel matched filter Communication Technology Laboratory Wireless Communication Group

26
**Form-Invariant Basepulses**

Continuous time system model impulse modulator g(t) h(t) h*(-t) g*(-t) P S kT Symbol discrete system model Communication Technology Laboratory Wireless Communication Group

27
**Transmission of Digital Information III: Decoding**

Communication Technology Laboratory Wireless Communication Group

28
**Maximum Likelihood Decoder and Decision Regions**

With orthonormal basepulse vector the impulse correlation matrix becomes the identity matrix The decoder observes the decision vector and generates an estimate of the transmit symbol vector To minimize the probability of error the decoder selects the hypothesis, which has the minimum Euclidean distance to the decision vector (Maximum Likelihood (ML) decoder) decision regions in the signal space decoder Communication Technology Laboratory Wireless Communication Group

29
**Example: Error Performance of QPSK**

Decision regions Gray mapping (bit 1, bit 2) (0,1) (1,1) (0,0) (1,0) Communication Technology Laboratory Wireless Communication Group

30
**Fundamentals of Short Range Wireless: Outline**

Digital transmission and detection on the AWGN channel digital transmission systems equivalent baseband model digital modulation and ML-detection Fading channels fading channels diversity MIMO wireless Modulation schemes for frequency selective channels multicarrier modulation Orthogonal Frequency Division Multiplexing (OFDM) Communication Technology Laboratory Wireless Communication Group

31
**Fading I: Time Selective (Narrowband) Fading Channels**

Communication Technology Laboratory Wireless Communication Group

32
**Path Loss and Short Term Fading**

TX RX distance (log(x)) power [dB] urban 40 dB/dec rural 30 dB/dec free space 20 dB/dec distance (log(x)) Communication Technology Laboratory Wireless Communication Group 32

33
**Doppler Shift I: 1800 Angle of Arrival**

Received signal in complex passband notation For (small scale effects) we obtain the complex envelope of the receive signal depends only on the displacement In the spectral domain we obtain the (spatial) Doppler shift Communication Technology Laboratory Wireless Communication Group 33

34
**Doppler Shift II: Arbitrary Angle of Arrival**

Complex envelope of received signal in the spectral domain we obtain the spatial Doppler shift For a linear movement of the receiver the spatial variations translate linearly into equivalent temporal variations the corresponding frequency shift follows as Communication Technology Laboratory Wireless Communication Group 34

35
**Multipath Propagation and Fading**

Complex envelope of the received signal Due to the different frequency shifts of the components, the magnitude of the received signal varies with the displacement: fading Example: note the spaced zero crossings 0.5 1 Communication Technology Laboratory Wireless Communication Group 35

36
**Doppler Spectrum: Power Spectral Density of Fading Process**

PSD Scattering coefficients cn modelled as uncorrelated random variables with variance fading described as random process Power spectral density (PSD) of fading process for note the relation between Doppler shift fxD,n and the angle of arrival fxD infinite number of scatterers under average receive power constraint: continuous PSD of fading process Communication Technology Laboratory Wireless Communication Group 36

37
**Jake's Doppler Spectrum**

uniform continuous scattering around receiver Relation of angle of arrival and Dopper shift cumulative power distribution Cumulative power distribution versus frequency Power spectral density "Jake's Spectrum" Communication Technology Laboratory Wireless Communication Group

38
**Jake's Channel Model for Linear Movement**

Multiplicative fading Speed of movement: v complex white Gaussian noise process fD Communication Technology Laboratory Wireless Communication Group

39
**Fading II: Frequency Selective Fading**

Communication Technology Laboratory Wireless Communication Group

40
**Broadband Channel Measurement**

Channel measurement with a short impulse h(t) (broadband) All scatterers, which lead to a given path delay are located on an ellipse Typical received signal: Communication Technology Laboratory Wireless Communication Group 40

41
**Scattering Function Doppler shift fx S4 S3 S1 S3 S2 S1 S4 S2**

The scattering function describes the average power spectral density of the received signal as a function of Doppler shift fx and delay Communication Technology Laboratory Wireless Communication Group 41

42
**Doppler Spectrum of Narrowband System**

shift S4 S1 S3 Doppler Spectrum Scattering function 2nd order statistics of the spatio-temporal fading process a narrow band system can not resolve the multiple paths narrowband fading with Doppler spectrum rms Doppler spread with the mean Doppler shift Communication Technology Laboratory Wireless Communication Group 42

43
**Delay Power Spectrum of Broadband System**

Doppler shift S4 Scattering function 2nd order statistics of the spatio-temporal fading process Delay power spectrum rms delay spread with the mean delay S1 S3 Delay power spectrum Communication Technology Laboratory Wireless Communication Group 43

44
**Classification of Multipath Channels**

The signal bandwidth and the duration of the transmit burst determine the fading model flat: no significant variation over the interval of interest selective: varies significantly over the interval of interest Narrowband systems experience frequency-flat fading Broadband systems experience frequency-selective fading A block fading model is suitable in the time-flat regime may be either frequency-flat or frequency selective Systems below the red curve are not physically implementable Note the role of Doppler spread and delay spread burst duration TBURST time-selective frequency-flat time-selective frequency-selective time-flat frequency-flat time-flat frequency-selective signal bandwidth B TBurst=1/B Communication Technology Laboratory Wireless Communication Group

45
**Typical Time-Selective/Frequency Selective Channel Model**

Structure Specification delay power spectrum delay + A discrete delay power spectrum is specified paths (delays) are usually clustered For each path (k) (i.e. delay ) a Doppler spectrum is specified default: Jake's spectrum if specified in terms of spatial frequency fx, substitute f x = f/v for linear movement with velocity v The filter coefficients are filtered complex normal random processes in line of sight (LoS) situations: nonzero mean Generation of fading processes white complex normal random process Communication Technology Laboratory Wireless Communication Group 45

46
**Special Cases + Block fading channel Frequency-flat fading**

Note that the coefficients are random variables (not processes) For each channel realization a new set of random variables is generated non LoS: Generation of fading processes white complex normal random process Communication Technology Laboratory Wireless Communication Group 46

47
**Fading III: Impact on Error Performance**

Communication Technology Laboratory Wireless Communication Group 47

48
**Frequency-Flat Fading Channel**

matched "filter" fading channel Symbol discrete system model with block fading block fading: fading variable instead of fading process note multiplication with magnitude of fading variable due to channel matched "filter" normalization of decision vector Communication Technology Laboratory Wireless Communication Group

49
**Error Performance of QPSK in Frequency-Flat Block Fading**

In frequency-flat block fading the error performance of QPSK is determined by the instantaneous value of the fading variable We can define various figures of merit. Frequently used are outage probability: probability, that the instantaneous bit error probability is above a target value fading averaged bit error probability Clearly these figures of merit depend on the probability density function (pdf) of the fading amplitude fading averaged bit error probability here is a chi2 random variable with 2 degrees of freedom Communication Technology Laboratory Wireless Communication Group

50
**Diversity Special case L=1 General case L=N: N-fold diversity**

the fading variable z is complex normally distributed; is the sum of two statistically independent squared real-valued normal random variable If , is Rayleigh distributed; Rayleigh fading if , is Rician distributed. K-factor: General case L=N: N-fold diversity For , is the sum of 2L squared real-valued Gaussian random variables chi2-distribution with 2L degrees of freedom e.g. achievable with L receive antennas Approximation: BER (c/SNR)L fading averaged bit error probability Communication Technology Laboratory Wireless Communication Group

51
**Fundamentals of Short Range Wireless: Outline**

Digital transmission and detection on the AWGN channel digital transmission systems equivalent baseband model digital modulation and ML-detection Fading channels fading channels diversity MIMO wireless Modulation schemes for frequency selective channels multicarrier modulation Orthogonal Frequency Division Multiplexing (OFDM) Communication Technology Laboratory Wireless Communication Group

52
**Vector/Matrix Channels**

Single Input/Single Output (SISO) channel coefficient Single Input/Multiple Output (SIMO) channel vector Multiple Input/Multiple Output (MIMO) channel matrix 52

53
Diversity Techniques Wireless channel varies in time, frequency and space Time, frequency and space diversity available Examples: Time diversity: repeat same codeword after channel varied (Repetition Code) Frequency diversity: transmit same symbol over two or more OFDM sub-carriers (if fading of the sub-carriers is uncorrelated) Space diversity: use more than one antenna at RX or TX or on both sides (But usually pure repetition is not an efficient way to code: repetition of the same information in time or frequency sacrifices bandwidth space diversity seems promising) 53 Diversity, MIMO

54
**Spatial (or Antenna) Diversity**

TX RX Receive diversity: using NRX receive antennas (spatial dimension) Transmit diversity: in addition a temporal coding needed Space-Time Codes High diversity factors available for high carrier frequencies and large bandwidths 54 Diversity, MIMO

55
**System Model with RX Diversity and Maximum Ratio Combining**

block fading vector channel channel matched "filter" Scalar symbol discrete system model note multiplication with magnitude of fading vector due to channel matched "filter" normalization of decision vector Communication Technology Laboratory Wireless Communication Group

56
Probability of Error TX RX Receive diversity: hi : channel gain between the TX antenna and the RX antenna i new argument of Q-Function: 56 Diversity, MIMO

57
**Array Gain and Diversity Gain**

Array (Power) gain Diversity gain 3 dB gain per doubling of the number of RX antenna Expression converges to constant for increasing NRX i.e. fading is eliminated in the limit 57

58
**Multiple Input/ Multiple Output**

Single Input/Single Output (SISO) channel coefficient Single Input/Multiple Output (SIMO) channel vector Multiple Input/Multiple Output (MIMO) channel matrix 58

59
**Free Space vs. Multipath Propagation**

scattering fading 59

60
**Multiple Antennas and Spatial Multiplexing**

Channel Matrix Singular Value Decomposition unitary full rank 3 spatial subchannels spatial multiplexing rank 1 60

61
**MIMO Wireless Capacity (1)**

TX RX MIMO channel capacity grows nearly linearly with N = min(NTX, NRX) [Foschini, Gans, 1998] [Telatar, 1999] K-factor of Rician fading Higher data rate without need of higher bandwidth spectral efficiency N decoupled spatial sub-channels available (Spatial Multiplexing) 61 Diversity, MIMO

62
**MIMO Wireless Capacity (2)**

TX RX K-factor of Rician fading Telatar, Foschini: NTX: number of TX antennas, NRX: number of RX antennas. 62 Diversity, MIMO

63
**MIMO Systems: Spatial Subchannels**

A priori transmit channel state information (CSIT) necessary ! Subchannels TX Diversity: Take only the “best“ subchannel ! Spatial Multiplexing: Take all ! SVD of MIMO channel matrix: 63 Diversity, MIMO

64
**MIMO Systems without CSIT: Spatial Subchannels**

Receiver has to compensate ISI due to V (cf. BLAST); if no a priori CSIT: TX Combining not possible; Spatial multiplexing leads to ISI 64 Diversity, MIMO

65
BLAST Architecture [Gesbert, et al.: From Theory to Practice: An Overview of MIMO Space–Time Coded Wireless Systems] 65 Diversity, MIMO

66
**BLAST (2) Antenna array at TX and RX**

Spatial Data Pipes in rich scattering (MIMO channel H of high rank) without increasing the bandwidth Spatial Multiplexing achieves ergodic MIMO capacity ISI compensation at RX necessary, because no CSIT used in BLAST system BLAST: no combination of diversity techniques and spatial multiplexing 66 Diversity, MIMO

67
**Fundamentals of Short Range Wireless: Outline**

Digital transmission and detection on the AWGN channel digital transmission systems equivalent baseband model digital modulation and ML-detection Fading channels fading channels diversity MIMO wireless Modulation schemes for frequency selective channels multicarrier modulation Orthogonal Frequency Division Multiplexing (OFDM) Communication Technology Laboratory Wireless Communication Group

68
**Multicarrier Modulation I: Continuous Time Implementation**

Communication Technology Laboratory Wireless Communication Group

69
**Discrete Channel Impulse Response**

kT g(t) h(t) h*(-t) g*(-t) S P -T -T T T p-1 p0 p1 + Communication Technology Laboratory Wireless Communication Group

70
**Low and High Data Rate Systems**

transmit basepulse channel impulse response t t t low data rate: no ISI high data rate: severe ISI Communication Technology Laboratory Wireless Communication Group

71
**Multicarrier Modulation**

H(f) Transmit in N subbands, for each of which the channel transfer function is approximately constant minimizes ISI in each subband subband center frequency fk for non-overlapping subbands, there is no inter-subband (inter-carrier) interference (ICI) One multicarrier (MC) symbol consists of N transmit symbols: Subbands implemented by letting The symbol rate on each subcarrier is f Communication Technology Laboratory Wireless Communication Group

72
**Transmit and Receive Window**

h(t) For sinc-windows there is strictly no interference between adjacent subcarriers (non-overlapping spectra) the ISI matrix in the discrete system model is diagonal Due to their infinite duration sinc-windows are not implementable. Is it possible to design finite length windows without introducing interference? Communication Technology Laboratory Wireless Communication Group

73
**Eigenfunction of the Convolution**

We consider the response of the channel h(t) to a step function with center frequency fk We observe a transient with duration Th After the transient the response is a scaled version of the input signal scaling factor H(fk) complex exponentials are eigenfunctions of convolution After an appropriate window, which blanks the transient, we obtain the input-output relation This observation is the key to the design of a finite window for MC t t t transient t Communication Technology Laboratory Wireless Communication Group

74
**Equivalent Diagonal Channel Matrix**

h(t) TX and RX window: Equivalent model: equivalent channel matrix Communication Technology Laboratory Wireless Communication Group

75
Receive Window As the equivalent channel matrix is diagonal, it does not affect orthogonality any more What is the impact of the receive window? Pulse correlation matrix the Fourier transform PRX(f) of the receive window pRX(t) determines the pulse correlation matrix For a uniform subcarrier spacing we have the Fourier transform of the receive window needs to have spaced zeros In this case the received window has to fullfil the following condition (temporal dual to the spectral Nyquist condition) the most compact window thus is given by this implies A temporal roll off improves the robustness to frequency offsets Communication Technology Laboratory Wireless Communication Group

76
**Transmit Window With we obtain Example for N=21; pTX(t)=rect(t/(Tc+Th)**

The transmit window has to be constant for at least This window determines the power spectral density of the transmit signal Communication Technology Laboratory Wireless Communication Group

77
Multicarrier Modulation II: Discrete Implementation (Orthogonal Frequency Division Multiplexing; OFDM) Communication Technology Laboratory Wireless Communication Group

78
**Discrete Channel Impulse Response**

kT g(t) h(t) h*(-t) g*(-t) P S Vector signal model -T -T T T p-1 p0 p1 + Communication Technology Laboratory Wireless Communication Group

79
**Response to Periodic Input Sequence**

Discrete channel impulse response with L taps for illustration assumed causal T T p0 p1 + Idea: use periodic input sequence to generate periodic response d dN d dN Equivalent channel matrix is circulant Communication Technology Laboratory Wireless Communication Group

80
**Cyclic Prefix and Circulant Channel Matrix**

Discrete channel impulse response with L taps for illustration assumed causal T T p0 p1 + NCP cyclic prefix d dN Circulant channel matrix due to cyclic prefix of length larger or equal to L-1 Communication Technology Laboratory Wireless Communication Group

81
**Diagonalization of Circulant Matrix: Orthogonal Frequency Division Multiplexing (OFDM)**

kT; k=1.. N+NCP kT; k=1.. N+NCP insert CP P S g(t) h(t) a(t) P S remove CP circulant channel matrix C diagonal channel matrix All (NxN) circulant matrices are diagonalized by the Fourier matrix FN Communication Technology Laboratory Wireless Communication Group

82
**Comparison of Discrete and Continuous Time Implementation of Multicarrier Modulation**

kT; k=1.. N+NCP kT; k=1.. N+NCP insert CP P S Low Pass Low Pass P S remove CP Note: Multicarrier Transmitter Multicarrier Receiver Communication Technology Laboratory Wireless Communication Group

Similar presentations

Presentation is loading. Please wait....

OK

CHAPTER 5 SIGNAL SPACE ANALYSIS

CHAPTER 5 SIGNAL SPACE ANALYSIS

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on conservation of environment poster Ppt on moral education Cardiopulmonary anatomy and physiology ppt on cells Ppt on atm machine software Ppt on recent natural disasters Ppt on time response analysis control Ppt on needle stick injury health Ppt on indian flag history Ppt on content addressable memory Download ppt on folk dances of india