In digital transmission, the number of bit errors is the number of received bits of a data stream over a communication channel that have been altered due to noise, interference, distortion or bit synchronization errors. The bit error rate (BER) is the number of bit errors per unit time. The bit error ratio (also BER) is the number of bit errors divided by the total number of transferred bits during a studied time interval. Bit error ratio is a unitless performance measure, often expressed as a percentage. The bit error probability pe is the expected value of the bit error ratio. The bit error ratio can be considered as an approximate estimate of the bit error probability. This estimate is accurate for a long time interval and a high number of bit errors. Example As an example, assume this transmitted bit sequence: 1 1 0 0 0 1 0 1 1 and the following received bit sequence: 0 1 0 1 0 1 0 0 1, The number of bit errors (the underlined bits) is, in this case, 3. The BER is 3 incorrect bits divided by 9 transferred bits, resulting in a BER of 0.333 or 33.3%. Packet error ratio The packet error ratio (PER) is the number of incorrectly received data packets divided by the total number of received packets. A packet is declared incorrect if at least one bit is erroneous. The expectation value of the PER is denoted packet error probability pp, which for a data packet length of N bits can be expressed as , assuming that the bit errors are independent of each other. For small bit error probabilities and large data packets, this is approximately Similar measurements can be carried out for the transmission of frames, blocks, or symbols. The above expression can be rearranged to express the corresponding BER (pe) as a function of the PER (pp) and the data packet length N in bits: Factors affecting the BER In a communication system, the receiver side BER may be affected by transmission channel noise, interference, distortion, bit synchronization problems, attenuation, wireless multipath fading, etc. The BER may be improved by choosing a strong signal strength (unless this causes cross-talk and more bit errors), by choosing a slow and robust modulation scheme or line coding scheme, and by applying channel coding schemes such as redundant forward error correction codes. The transmission BER is