Friday, September 12, 2014

Wireless Communication: Overview


The practice of wireless communication is not new. We have been using this principle from the ages without understanding the basics. Wireless communication is the act of transmitting and receiving information or signal via the air medium, which actually uses the electromagnetic waves as a means of transmission. When we talk to somebody, we are inherently using the wireless communication. 

Since we can talk to people nearby, why cant we talk to the people far away? If we have enough power to speak very loudly then we can even talk to the people at some distance but not too far. After the development of radio and electromagnetic waves, scientists knew that the information can be transmitted in the form of electromagnetic waves and the challenge was to determine the ways to transmit these signals. Later, scientists like Shannon founded the idea that the information can be measured and this led to the development of whole lot of aspects in the information theory and wireless communications.

The deeper knowledge of the electromagnetics and antenna theory and the also the development in the field of digital electronics made it possible to digitize the earlier form of analog communication, which was limited only to the voice transmission (1G). After then a whole lot of development has been observed in terms of the data transmission and the quality of service of the wireless users. 2G with SMS facilities, 2.5G with GPRS and 3G with high speed data connection and improving. 

The latest challenge on wireless communication is to meet the incresing demands of the people in terms of the data rate and the quality of service with the limited spectrum and limited power available. To meet these demands, different multiplexing and modulation techniques were proposed in the past. Orthogonal Frequency Division Multiplexing (OFDM) is recently used multi-carrier modulation technique, used in Long Term Evolution (LTE) standard, where instead of single wide band carrier, the data is transmitted over multiple parallel orthogonal narrow-band carriers.

However, researches are still going on to meet the increasing data requirements with the limited power constraint and the limited spectrum. The concept of interference channel (IC) is not new. This is talked from the time of Shannon. But, it is still an open problem to characterize the capacity of even a two user IC. A number of research works are available in this regard.

Since all the transmitters transmit at the same time using all the available bandwidth, each receiver receives the interference from all other unwanted transmitters and this interference is the limiting factor in IC and a major part of the latest research on wireless communication is focused in managing the interference so that better data rate and the quality of service is achieved.

The capacity of a multi-user interference channel is expressed in terms of the signal to interference plus noise ratio (SINR) as given by the Shannon's channel capacity formula:
                                               $C=d \log_2 (1+\text{SINR})$,
where $d$ is the prelog factor also called the degrees of freedom (DoF). In other words, DoF can be defined as the number of data streams that can be transmitted without interfering the other receivers. In multiple input multiple output (MIMO) systems, DoF is the number of interference free parallel channels between a transmitter and a receiver.

As we see that the capacity increases linearly with the DoF (d), it is a good idea to improve the DoF of the system instead of giving much attention to improve the SINR which is just growing logarithmically. Interference alignment (IA), the latest developed interference management technique, aims at improving the DoF of a multi-user interfering system by aligning all the interference in a common subspace and thus providing enough space for the desired data streams to be decoded. IA is believed to provide the outerbound on the achievable DoF, which is considered to be $\frac{K}{2}$ for a $K$-user single input single output (SISO) interference channel. Number of references are available in this regard.
                                         



Tuesday, September 9, 2014

Multiplication by 9, 99, 999 , 9999 and so on

Actually, multiplication by 9, 99, 999, 9999 and so on series seems easy and simple. This can be achieved by just two operation of subtraction and addition by shifting the positions to the right equal to the number of 9 used to multiply.

The rule that works for all the cases is 

Subtract (multiplicand-1) from multiplier (which is the series of 9), then add the resultant with (multiplicand-1) by shifting the 'n' positions to the right, where 'n' is the minimum number of   multiplier or multiplicand, n=min(no.  of multiplicand digits, no. of multiplier digits) in the series. 

Consider a simple example:

$8 \times 9=72$,

  1.  Multiplicand-1 = 8-1=7,
  2.  Subtract 7 from 9, 9-7=2,
  3. Shift 2 to the right one position and add one zero after 7 and perform bit-wise addition  as 

                                             02
                                             70
                                                          
                                             72
$88 \times 9=792$
  1.  Multiplicand-1 = 88-1=87,
  2.  Subtract 87 from 9, 9-87=-82,
  3. Shift -82 to the right one position and add with 87  i.e 
                                       (-08) 2
                                          87  0
                                                     
                                     (79)    2
$888 \times 9=7992$
  1.  Multiplicand-1 = 888-1=887,
  2.  Subtract 887 from 9, 9-887=-882,
  3. Shift -882 to the right one position and add with 887  i.e 
                                     (-088) 2
                                        887  0
                                                    
                                    (799)   2
$8 \times 99=792$
       

  1. Multiplicand-1= 8-1=7,
  2. Subtract 7 from 99 = 99-7=92,
  3. Shift 92 to the right by one and perform the addition as  
                               700
                               092
                                             
                               792