08-08-2012, 11:09 AM
Short notes on OFDM
[attachment=30493]
The passband bandwidth required to transmit at a rate Rb is at least Rb and can go up to 2Rb for a RC
pulse shape with a = 1. When we want to transmit at a large rate we will need proportionally larger
bandwidth.
Physical channels are typically well behaved over small windows of frequency – typically referred
to as the Coherent Bandwidth. Over this bandwidth, the response of the channel is reasonably flat
and hence the frequency components of the signal add in-phase (coherently) and do not get
distorted. When the transmission bandwidth is larger than the coherence bandwidth of the channel,
we need to employ equalizers to 'undo' the effect of the channel. This is mostly cumbersome as it
involves additional blocks/resources in the system to estimate the channel and to compensate for the
distortion introduced by the channel. Presence of noise causes additional degradation and if the
channels has spectral nulls/dips noise variance increases causing noise enhancement.
One way to overcome this, without a penalty on bandwidth, is to split the input data stream into
parallel streams of lower rate data and transmit them over different 'channels' in a frequency
division multiplexed fashion. The most efficient way to do it is to 'pack' the channels as close to
each other as possible – yet at the same time keep the channels orthogonal (independent). This is
achieved by OFDM (Orthogonal Frequency Division Multiplexing).
Inter-Carrier Interference (ICI)
Thus OFDM consists of multiple narrow band channels (at carrier frequencies f0, f1, f2, .. ,fN-1) with
Rb as the inter-carrier spacing. The spectrum of OFDM is hence composed of multiple narrow band
signals each centered around the carrier frequencies f0, f1, f2, .. ,fN-1. As we want the carriers to be
orthogonal, we have to carefully choose the time domain pulse shape for maintaining this
orthogonality. The loss of orthogonality in frequency domain will lead to the data on different subcarriers
interference with each other, and this effect is referred to as Inter Carrier Interference (ICI).
This effect is the frequency domain equivalent for Inter-Symbol Interference and the dual
relationship of time and frequency domain will help us find good waveforms for minimizing ICI.
For now we will use the rectangular window w(t) defined as below and later come up with a better
time domain window in later sections.
Modulation and demodulation of OFDM signals
Although the process of converting from single frequency operation to a multi frequency operation
might suggest the use of multiple oscillators to generate the individual sub-carriers, the process of
modulation and demodulation are achieved efficiently using IFFT and FFT operators. Historically,
advances in DFT and FFT enabled OFDM to be popular, even though the underlying principles
were know much earlier.
The actual process of OFDM transmission consists of coming up with the weighting values of the N
sub-carriers which are derived from appropriate signal constellation (most popularly BPSK, QPSK,
QAM and the differential PSKs). The combined frequency domain signal is converted into time
domain using efficient IFFT operations to produce N time domain samples. This is one of the
reason why N is usually a power of 2.
[attachment=30493]
The passband bandwidth required to transmit at a rate Rb is at least Rb and can go up to 2Rb for a RC
pulse shape with a = 1. When we want to transmit at a large rate we will need proportionally larger
bandwidth.
Physical channels are typically well behaved over small windows of frequency – typically referred
to as the Coherent Bandwidth. Over this bandwidth, the response of the channel is reasonably flat
and hence the frequency components of the signal add in-phase (coherently) and do not get
distorted. When the transmission bandwidth is larger than the coherence bandwidth of the channel,
we need to employ equalizers to 'undo' the effect of the channel. This is mostly cumbersome as it
involves additional blocks/resources in the system to estimate the channel and to compensate for the
distortion introduced by the channel. Presence of noise causes additional degradation and if the
channels has spectral nulls/dips noise variance increases causing noise enhancement.
One way to overcome this, without a penalty on bandwidth, is to split the input data stream into
parallel streams of lower rate data and transmit them over different 'channels' in a frequency
division multiplexed fashion. The most efficient way to do it is to 'pack' the channels as close to
each other as possible – yet at the same time keep the channels orthogonal (independent). This is
achieved by OFDM (Orthogonal Frequency Division Multiplexing).
Inter-Carrier Interference (ICI)
Thus OFDM consists of multiple narrow band channels (at carrier frequencies f0, f1, f2, .. ,fN-1) with
Rb as the inter-carrier spacing. The spectrum of OFDM is hence composed of multiple narrow band
signals each centered around the carrier frequencies f0, f1, f2, .. ,fN-1. As we want the carriers to be
orthogonal, we have to carefully choose the time domain pulse shape for maintaining this
orthogonality. The loss of orthogonality in frequency domain will lead to the data on different subcarriers
interference with each other, and this effect is referred to as Inter Carrier Interference (ICI).
This effect is the frequency domain equivalent for Inter-Symbol Interference and the dual
relationship of time and frequency domain will help us find good waveforms for minimizing ICI.
For now we will use the rectangular window w(t) defined as below and later come up with a better
time domain window in later sections.
Modulation and demodulation of OFDM signals
Although the process of converting from single frequency operation to a multi frequency operation
might suggest the use of multiple oscillators to generate the individual sub-carriers, the process of
modulation and demodulation are achieved efficiently using IFFT and FFT operators. Historically,
advances in DFT and FFT enabled OFDM to be popular, even though the underlying principles
were know much earlier.
The actual process of OFDM transmission consists of coming up with the weighting values of the N
sub-carriers which are derived from appropriate signal constellation (most popularly BPSK, QPSK,
QAM and the differential PSKs). The combined frequency domain signal is converted into time
domain using efficient IFFT operations to produce N time domain samples. This is one of the
reason why N is usually a power of 2.