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In GMSK modulation (used in GSM and DECT standard), a GMSK signal is generated by shaping the information bits in NRZ format through a Gaussian Filter. The filtered pulses are then frequency modulated to yield the GMSK signal. GMSK modulation is quite insensitive to non-linearities of power amplifier and is robust to fading effects. But it has a moderate spectral efficiency.
An expression for the Gaussian Filter with 3dB Bandwidth is derived here.
The requirements for a gaussian filter used for GMSK modulation in GSM/DECT standard are as follows,
Now the challenge is to design a Gaussian Filter fG(t) that satifies the 3dB bandwidth requirement i.e. in the frequency domain at some frequency f=B, the filter should posses -3dB gain ( in otherwords => half power point located at f=B)
The expression for the required Gaussian Filter can be obtained by choosing the variance of the above mentioned distribution so that the Fourier Transform of the above mentioned expression has a -3dB power gain at f=B.
The fourier transform of the above mentioned expression is
A signal is a bandpass signal if we can fit all its frequency content inside a bandwidth . Bandwidth is simply the difference between the lowest and the highest frequency present in the signal.
“In order for a faithful reproduction and reconstruction of a bandpass analog signal with bandwidth – , the signal should be sampled at a Sampling frequency () that is greater than or equal to twice the maximum bandwidth of the signal:
Consider a bandpass signal extending from 150Hz to 200Hz. The bandwidth of this signal is . In order to faithfully represent the above signal in the digital domain the sampling frequency must be . Note that the sampling frequency 100Hz is far below the maximum content of the signal (which is 200Hz). That is why the bandpass sampling is also called “under-sampling”. As long as the sampling frequency is greater than or equal to twice the bandwidth of the signal, the reconstruction back to analog domain will be error free.
Going back to the aliasing zone figure, if the signal of interest is in the zone other than zone 1, it is called a bandpass signal and the sampling operation is called “Intermediate Sampling” or “Harmonic Sampling” or “Under Sampling” or “Bandpass Sampling”.
Folding Frequencies and Aliasing Zones
Note that zone 1 is a mirror image of zone 2 (with frequency reversal). Similarly zone 3 is a mirror image of zone 4 etc.., Also, any signal in zone 1 will be reflected in zone 2 with frequency reversal which inturn will be copied in zone 3 and so on.
Lets say the signal of interest lies in zone 2. This will be copied in all the other zones. Zone 1 also contains the sampled signal with frequency reversal which can be correct by reversing the order of FFT in digital domain.
No matter in which zone the signal of interest lies, zone 1 always contains the signal after sampling operation is performed. If the signal of interest lies in any of the even zones, zone 1 contains the sampled signal with frequency reversal. If the signal of interest lies in any of the odd zones, zone 1 contains the sampled signal without frequency reversal.
Example:
Consider an AM signal centered at carrier frequency 1MHz, with two components offset by 10KHz – 0.99 MHz and 1.01 MHz. So the AM signal contains three frequency components at 0.99 MHz, 1 MHz and 1.01 MHz.
Our desire is to sample the AM signal. As with the usual sampling theorem (baseband), we know that if we sample the signal at twice the maximum frequency i.e Fs>=2*1.01MHz=2.02 MHz there should be no problem in representing the analog signal in digital domain.
By the bandpass sampling theorem, we do not need to use a sampler running at Fs>=2.02 MHz. Faster sampler implies more cost. By applying the bandpass sampling theorem, we can use a slower sampler and reduce the cost of the system. The bandwidth of the signal is 1.01MHz-0.99 MHz = 20 KHz. So, just sampling at will convert the signal to digital domain properly and we can also avoid using an expensive high rate sampler (if used according to baseband sampling theorem).
Lets set the sampling frequency to be (which is 3 times higher than the minimum required sampling rate of 40KHz or oversampling rate =3).
Now we can easily find the position of the spectral components in the sampled output by using the aliasing zone figure as given above. Since , will be 60KHz. So the zone 1 will be from 0 to 60 KHz, zone 2 -> 60-120KHz and so on. The three spectral components at 0.99MHz, 1MHz and 1.01 MHz will fall at zone 17 ( how ? 0.99 MHz/60 KHz = 16.5 , 1MHz/60KHz = 16.67 and 1.01MHz/60KHz = 16.83 , all figures approximating to 17). By the aliasing zone figure, zone 16 contains a copy of zone 17, zone 15 contains a copy of zone 16, zone 14 contains a copy of zone 15 and so on… Finally zone 1 contains the copy of zone 2 (Frequency reversal also exist at even zones). In effect, zone 1 contains a copy of zone 17. Since the original spectral components are at zone 17, which is an odd zone, zone 1 contains the copy of spectral components at zone 17 without frequency reversal.
Since there is no frequency reversal, in zone 1 the three components will be at 30KHz, 40KHz and 50KHz (You can easily figure this out ).
This operation has downconverted our signal from zone 17 to zone 1 without distorting the signal components. The downconverted signal can be further processed by using a filter to select the baseband downconverted components. Following figure illustrates the concept of bandpass sampling.
Bandpass Sampling
Consider the same AM signal with three components at 0.99MHz, 1MHz and 1.01 MHz. Now we also have an “unwanted” fourth component at 1.2 MHz along with the incoming signal. If we sample the signal at 120 KHz, it will cause aliasing (because the bandwidth of the entire signal is 1.2-0.99 = 0.21 MHz = 210 KHz and the sampling frequency of 120KHz is below twice the bandwidth). In order to avoid anti-aliasing and to discard the unwanted component at 1.2 MHz, an anti-aliasing bandpass filter has to be used to select those desired component before performing the sampling operation at 120KHz. This is also called “pre-filtering”. The following figure illustrates this concept.
“Nyquist-Shannon Sampling Theorem” is the fundamental base over which all the digital processing techniques are built. Processing a signal in digital domain gives several advantages (like immunity to temperature drift, accuracy, predictability, ease of design, ease of implementation etc..,) over analog domain processing.
Analog to Digital conversion:
In analog domain, the signal that is of concern is continuous in both time and amplitude. The process of discretization of the analog signal in both time domain and amplitude levels yields the equivalent digital signal. The conversion of analog to digital domain is a three step process 1) Discretization in time – Sampling 2) Discretization of amplitude levels – Quantization 3) Converting the discrete samples to digital samples – Coding/Encoding
Components of ADC
The sampling operation samples (“chops”) the incoming signal at regular interval called “Sampling Rate” (denoted by ). Sampling Rate is determined by Sampling Frequency (denoted by ) as Lets consider the following logical questions: * Given a real world signal, how do we select the sampling rate in order to faithfully represent the signal in digital domain ? * Is there any criteria for selecting the sampling rate ? * Will there be any deviation if the signal is converted back to analog domain ? Answer : Consult the “Nyquist-Shannon Sampling Theorem” to select the sampling rate or sampling frequency.
Nyquist-Shannon Sampling Theorem:
The following sampling theorem is the exact reproduction of text from Shannon’s classic paper[1],
“If a function f(t) contains no frequencies higher than W cps, it is completely determined by giving its ordinates at a series of points spaced 1/2W seconds apart.”
Sampling Theorem mainly falls into two categories : 1) Baseband Sampling – Applied for signals in the baseband (useful frequency components extending from 0Hz to some Fm Hz) 2) Bandpass Sampling – Applied for signals whose frequency components extent from some F1 Hz to F2Hz (where F2>F1) In simple terms, the Nyquist Shannon Sampling Theorem for Baseband can be explained as follows
Baseband Sampling:
If the signal is confined to a maximum frequency of Fm Hz, in other words, the signal is a baseband signal (extending from 0 Hz to maximum Fm Hz).
In order for a faithful reproduction and reconstruction of an analog signal that is confined to a maximum frequency Fm, the signal should be sampled at a Sampling frequency (Fs) that is greater than or equal to twice the maximum frequency of the signal:
Consider a 10Hz sine wave in analog domain. The maximum frequency present in this signal is Fm=10 Hz (obviously no doubt about it !!!). Now, to satisfy the sampling theorem that is stated above and to have a faithful representation of the signal in digital domain, the sampling frequency can be chosen as Fs >=20Hz. That is, we are free to choose any number above 20 Hz. Higher the sampling frequency higher is the accuracy of representation of the signal. Higher sampling frequency also implies more samples, which implies more storage space or more memory requirements. In time domain, the process of sampling can be viewed as multiplying the signal with a series of pulses (“pulse train) at regular interval – Ts. In frequency domain, the output of the sampling process gives the following components – Fm (original frequency content of the signal), Fs±Fm,2Fs±Fm,3Fs±Fm,4Fs±Fm and so on and on…
Baseband Sampling
Now the sampled signal contains lots of unwanted frequency components (Fs±Fm,2Fs±Fm,…). If we want to convert the sampled signal back to analog domain, all we need to do is to filter out those unwanted frequency components by using a “reconstruction” filter (In this case it is a low pass filter) that is designed to select only those frequency components that are upto Fm Hz. The above process mentions only the sampling part which samples the incoming analog signal at regular intervals. Actually a quantizer will follow the sampler which will discretize (“quantize”) amplitude levels of the sampled signal. The quantized amplitude levels are sent to an encoder that converts the discrete amplitude levels to binary representation (binary data). So when converting the binary data back to analog domain, we need a Digital to Analog Converter (DAC) that converts the binary data to analog signal. Now the converted signal after the DAC contains the same unwanted frequencies as well as the wanted component. Thus a reconstruction filter with proper cut-off frequency has to placed after the DAC to filter out only the wanted components.
Aliasing and Anti-aliasing:
Consider a signal with two frequency components f1=10Hz – which is our desired signal and f2=20Hz – which is a noise. Let’s say we sample the signal at 30Hz. The first frequency component f1=10Hz will generate following frequency components at the output of the multiplier (sampler) – 10Hz,20Hz,40Hz,50Hz,70Hz and so on. The second frequency component f2=20Hz will generate the following frequency components at the output of the multiplier – 20Hz,10Hz,50Hz,40Hz,80Hz and so on…
Aliasing and Anti-aliasing
Note the 10Hz component that is generated by f2=20Hz. This 10Hz component (which is a manifestation of noisy component f2=20Hz) will interfere with our original f1=10Hz component and are indistinguishable. This 10Hz component is called “alias” of the original component f2=20Hz (noise). Similarly the 20Hz component generated by f1=10Hz component is an “alias” of f1=10Hz component. This 20Hz alias of f1=10Hz will interfere with our original component f2=20Hz and are indistinguishable. We do not need to care about the interference that occurs at 20Hz since it is a noise and any way it has to be eliminated. But we need to do something about the aliasing component generated by the f2=20Hz. Since this is a noise component, the aliasing component generated by this noise will interfere with our original f1=10Hz component and will corrupt it. Aliasing depends on the sampling frequency and its relationship with the frequency components. If we sample a signal at Fs, all the frequency components from Fs/2 to Fs will be alias of frequency components from 0 to Fs/2 and vice versa. This frequency – Fs/2 is called “Folding frequency” since the frequency components from Fs/2 to Fs folds back itself and interferes with the components from 0Hz to Fs/2 Hz and vice versa. Actually the aliasing zones occur on the either sides of 0.5Fs, 1.5Fs, 2.5Fs,3.5Fs etc… All these frequencies are also called “Folding Frequencies” that causes frequency reversal. Similarly aliasing also occurs on either side of Fs,2Fs,3Fs,4Fs… without frequency reversals. The following figure illustrates the concept of aliasing zones.
Folding Frequencies and Aliasing Zones
In the above figure, zone 2 is just a mirror image of zone 1 with frequency reversal. Similarly zone 2 will create aliases in zone 3 (without frequency reversal), zone 3 creates mirror image in zone 4 with frequency reversal and so on… In the example above, the folding frequency was at Fs/2=15Hz, so all the components from 15Hz to 30Hz will be the alias of the components from 0Hz to 15Hz. Once the aliasing components enter our band of interest, it is impossible to distinguish between original components and aliased components and as a result, the original content of the signal will be lost. In order to prevent aliasing, it is necessary to remove those frequencies that are above Fs/2 before sampling the signal. This is achieved by using an “anti-aliasing” filter that precedes the analog to digital converter. An anti-aliasing filter is designed to restrict all the frequencies above the folding frequency Fs/2 and therefore avoids aliasing that may occur at the output of the multiplier otherwise.
A complete design of ADC and DAC
Thus, a complete design of analog to digital conversion contains an anti-aliasing filter preceding the ADC and the complete design of digital to analog conversion contains a reconstruction filter succeeding the DAC.
ADC and DAC chain
Note: Remember that both the anti-aliasing and reconstruction filters are analog filters since they operate on analog signal. So it is imperative that the sampling rate has to be chosen carefully to relax the requirements for the anti-aliasing and reconstruction filters.
Effects of Sampling Rate:
Consider a sinusoidal signal of frequency Fm=2MHz. Lets say that we sample the signal at Fs=8MHz (Fs>=2*Fm). The factor Fs/Fm is called “over-sampling factor”. In this case we are over-sampling the signal by a factor of Fm=8MHz/2MHz = 4. Now the folding frequency will be at Fs/2 = 4MHz. Our anti-aliasing filter has to be designed to strictly cut off all the frequencies above 4MHz to prevent aliasing. In practice, ideal brick wall response for filters is not possible. Any filter will have a transition band between pass-band and stop-band. Sharper/faster roll off transition band (or narrow transition band) filters are always desired. But such filters are always of high orders. Since both the anti-aliasing and reconstruction filters are analog filters, high order filters that provide faster roll-off transition bands are expensive (Cost increases proportionally with filter order). The system also gets bulkier with increase in filter order.Therefore, to build a relatively cheaper system, the filter requirement in-terms of width of the transition band has to be relaxed. This can be done by increasing the sampling rate or equivalently the over-sampling factor. When the sampling rate (Fs) is increased, the distance between the maximum frequency content Fm and Fs/2 will increase. This increase in the gap between the maximum frequency content of the signal and Fs/2 will ease requirements on the transition bands of the anti-aliasing analog filter. Following figure illustrates this concept. If we use a sampling frequency of Fs=8MHz (over-sampling factor = 4), the transition band is narrower and it calls for a higher order anti-aliasing filter (which will be very expensive and bulkier). If we increase the sampling frequency to Fs=32MHz (over-sampling factor = 32MHz/2MHz=16), the distance between the desired component and Fs/2 has greatly increased that it facilitates a relatively inexpensive anti-aliasing filter with a wider transition band. Thus, increasing the sampling rate of the ADC facilitates simpler lower order anti-aliasing filter as well as reconstruction filter. However, increase in the sampling rate calls for a faster sampler which makes ADC expensive. It is necessary to compromise and to strike balance between the sampling rate and the requirement of the anti-aliasing/reconstruction filter.
Effects of Sampling Rate
Application : Up-conversion
In the above examples, the reconstruction filter was conceived as a low pass filter that is designed to pass only the baseband frequency components after reconstruction. Remember that any frequency component present in zone 1 will be reflected in all the zones (with frequency reversals in even zones and without frequency reversals in odd zones). So, if we design the reconstruction filter to be a bandpass filter selection the reflected frequencies in any of the zones expect zone 1, then we achieve up-conversion. In any communication system, the digital signal that comes out of a digital signal processor cannot be transmitted across as such. The processed signal (which is in the digital domain) has to be converted to analog signal and the analog signal has to be translated to appropriate frequency of operation that fits the medium of transmission. For example, in an RF system, a baseband signal is converted to higher frequency (up-conversion) using a multiplier and oscillator and then the high frequency signal is transmitted across the medium. If we have a band-pass reconstruction filter at the output of the DAC, we can directly achieve up-conversion (which saves us from using a multiplier and oscillator). The following figure illustrates this concept.
Goal: Simulate discrete-time cyclic-prefixed OFDM communication system. Explain role of IFFT/FFT, cyclic prefix. Simulate M-QPSK / M-QAM based cyclic prefixed OFDM over AWGN channel.
The schematic diagram of a simplified cyclic-prefixed OFDM (CP-OFDM) data transmission system is shown in Figure 1. The basic parameter to describe an OFDM system is to specify the number of subchannels () required to send the data. The number of subchannels is typically set to powers of 2, such as . The size of inverse discrete Fourier transform (IDFT) and discrete Fourier transform (DFT) need to be set accordingly.
The transmission begins by converting the source information stream into parallel subchannels. For convenience, the information stream is already represented as a symbol from the set . The data symbol in each subchannel is modulated using the chosen modulation technique such as MPSK or MQAM.
Since this is a baseband discrete-time model, where the signals are represented at symbol sampling instants, the information symbol on each parallel stream is assumed to be modulating a single orthogonal carrier. At this juncture, the modulated symbols on the parallel streams can be visualized as coming from different orthogonal subchannels in the frequency domain. The components of the orthogonal subchannels in the frequency domain are converted to time domain using IDFT operation.
Figure 1: Discrete-time simulation model for OFDM transmission and reception
The following generic function implements the modulation mapper (constellation mapping) shown in the Figure 1. The function supports MPSK modulation for and MQAM modulation that has square constellation : . It is built over the mpsk_modulator.m and mqam_modulator.m functions given in sections 5.3.2 and 5.3.3 of chapter 5 (Refer the book Wireless communication systems using Matlab).
modulation_mapper.m: Implementing the modulation mapper for MPSK and MQAM
function [X,ref]=modulation_mapper(MOD_TYPE,M,d)
%Modulation mapper for OFDM transmitter
% MOD_TYPE - 'MPSK' or 'MQAM' modulation
% M - modulation order, For BPSK M=2, QPSK M=4, 256-QAM M=256 etc..,
% d - data symbols to be modulated drawn from the set {1,2,...,M}
%returns
% X - modulated symbols
% ref -ideal constellation points that could be used by IQ detector
if strcmpi(MOD_TYPE,'MPSK'),
[X,ref]=mpsk_modulator(M,d);%MPSK modulation
else
if strcmpi(MOD_TYPE,'MQAM'),
[X,ref]=mqam_modulator(M,d);%MQAM modulation
else
error('Invalid Modulation specified');
end
end;end
OFDM signal is a composite signal that contains information from subchannels. Since the modulated symbols are visualized to be in frequency domain, it is converted to time-domain using IDFT. In the receiver, the corresponding inverse operation is performed by the DFT block. The IDFT and DFT blocks in the schematic can also be interchanged and it has no impact to the transmission.
In a time-dispersive channel, the orthogonality of the subcarriers cannot be maintained in a perfect state due to delay distortion. This problem is addressed by adding a cyclic extension (also called cyclic prefix) to the OFDM symbol (reference [1]). A cyclic extension is added by copying the last symbols from the vector and pasting it to its front as shown in Figure 2.
Figure 2: Adding a cyclic prefix in CP-OFDM
Cyclic extension of OFDM symbol converts the linear convolution channel to a channel performing cyclic convolution (view demo here) and this ensures orthogonality of subcarriers in a time-dispersive channel. It also completely eliminates the subcarrier interference as long as the impulse response of the channel is shorter than the cyclic prefix. At the receiver, the added cyclic prefix is simply removed from the received OFDM symbol.
On the receiver side, the demapper for demodulating MPSK and MQAM can be implemented by using a simple IQ detector that uses the minimum euclidean distance metric for demodulation. (discussion and function definitions in section 5.4.4 of chapter 5 (Refer the book Wireless communication systems using Matlab).
Performance of MPSK-CP-OFDM and MQAM-CP-OFDM on AWGN channel
The code (given in the book Wireless communication systems using Matlab) puts together all the functional blocks of an OFDM transmission system, that were described here, to simulate the performance of a CP-OFDM system over an AWGN channel. The code supports two types of underlying modulations for OFDM – MPSK or MQAM. It generates random data symbols, modulates them using the chosen modulation type, converts the modulated symbols to frequency domain using IDFT operation and adds cyclic prefix to form an OFDM symbol. The resulting OFDM symbols are then added with AWGN noise vector that corresponds to the specified value (AWGN noise model is described in this article).
On the receiver side, cyclic prefix is removed from the received OFDM symbol, DFT is performed and then the symbols are sent through a demapper for getting an estimate of the source symbols. The demapper is implemented by using a simple IQ detector that uses the minimum euclidean distance metric for demodulation. Finally, the symbol error rates are computed and compared against the theoretical symbol error rate curves for the respective modulations over AWGN. Simulated performance results are plotted in Figure 3.
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Consider a non-ideal channel h(t)≠δ(t), that causes delay dispersion. Delay dispersion manifests itself as Inter Symbol Interference (ISI)on each subcarrier channel due to pulse overlapping. It will also cause ICC (Inter Carrier Interference ) due to the non-orthogonality of the received signal. Adding cyclic prefix to each OFDM symbol mitigates the problems of ISI and ICC by removing them altogether.
Lets say, without cyclic prefix we transmit the following N values (N=Nfft=length of FFT/IFFT) for a single OFDM symbol.
$$ X_0,X_1,X_2,…,X_{N-1} $$
Lets consider a cyclic prefix of length Ncp, ( where Ncp<N ), is formed by copying the last Ncp values from the above vector of X and adding those Ncp values to the front part of the same X vector.With a cyclic prefix length Ncp, ( where Ncp<N ), the following values constitute a single OFDM symbol :
If T is the duration of the an OFDM symbol in secs, due to the addition of cyclic prefix of length Ncp, the total duration of an OFDM symbol becomes T+Tcp, where Tcp=Ncp*T/N. Therefore, the number of samples allocated for cyclic prefix can be calculated as Ncp=Tcp*N/T, where N is the FFT/IFFT length, T is the IFFT/FFT period and Tcp is the duration of cyclic prefix.
The key ideas behind adding cyclic prefix :
1) Convert linear convolution in to circular convolution which eases the process of detecting the received signal by using a simple single tap equalizer
If you wish to know how the addition of cyclic prefix converts linear convolution to circular convolution, visit this link
2) Help combat ISI and ICC.
When a cyclic prefix of length Ncp is added to the OFDM symbol, the output of the channel (r) is given by circular convolution of channel impulse response (h) and the OFDM symbols with cyclic prefix (x).
$$ r=h \circledast x $$
As we know, for the discrete signals, circular convolution in the time domain translates to multiplication in the frequency domain.Thus, in frequency domain, the above equation translates to
$$ R=HX $$
At the receiver, R is the received signal (in Frequency domain) and our goal is to estimate the transmitted signal (X) from the received signal R. From the above equation, the problem of detecting the transmitted signal at the receiver side translate to a simple equalization equation as follows
$$ \hat{X}= \frac{R}{H} $$
After the FFT performed at the receiver side (i.e. after the FFT block in the receiver side), a single tap equalizer (which essentially implements the above equation) is used to estimate the transmitted OFDM symbol. It also corrects the phase and equalizes the amplitude.
A basic OFDM architecture with Cyclic Prefix is given below. (In the following diagram, symbols represented by small case letters are assumed to be in time domain, whereas the symbols represented by uppercase letters are assumedto be in frequency domain)
An OFDM communication Architecture with Cyclic Prefix
The IEEE specs specify the length of the Cyclic prefix in terms of its duration.
Lets see how to convert the specified duration (Tcp) in to actual number of samples assigned for cyclic prefix (Ncp).
Lets see an example of deriving Ncp from IEEE 802.11 spec [1]
Given parameters in the spec:
N=64; %FFT size or total number of subcarriers (used + unused) 64
Nsd = 48; %Number of data subcarriers 48
Nsp = 4 ; %Number of pilot subcarriers 4
ofdmBW = 20 * 10^6 ; % OFDM bandwidth
Derived Parameters:
deltaF = ofdmBW/N; % Bandwidth for each subcarrier - include all used and unused subcarries
Tfft = 1/deltaF; % IFFT or FFT period = 3.2us
Tgi = Tfft/4; % Guard interval duration - duration of cyclic prefix - 1/4th portion of OFDM symbols
Tsignal = Tgi+Tfft; % Total duration of BPSK-OFDM symbol = Guard time + FFT period
Ncp = N*Tgi/Tfft; %Number of symbols allocated to cyclic prefix
Nst = Nsd + Nsp; % Number of total used subcarriers
nBitsPerSym=Nst; %For BPSK the number of Bits per Symbol is same as number of subcarriers
In the previous article, the architecture of an OFDM transmitter was described using sinusoidal components. Generally, an OFDM signal can be represented as
\[OFDM\; signal = c(t)=\sum_{n=0}^{N-1}s_{n}(t)sin(2\pi f_{n}t )\]
\(s(t)\) = symbols mapped to chosen constellation (BPSK/QPSK/QAM etc..,)
\(f_n\) = orthogonal frequency
This equation can be thought of as an IFFT process ( Inverse Fast Fourier Transform). The Fourier transform breaks a signal into different frequency bins by multiplying the signal with a series of sinusoids. This essentially translates the signal from time domain to frequency domain. But, we always view IFFT as a conversion process from frequency domain to time domain.
FFT is represented by
\[X(k)=\sum_{n=0}^{N-1}x(n) \cdot sin \left(\frac{2\pi kn}{N} \right)+j\sum_{n=0}^{N-1}x(n) \cdot cos \left(\frac{2\pi kn}{N} \right)\]
\[x(n)=\sum_{n=0}^{N-1}X(k) \cdot sin \left(\frac{2\pi kn}{N} \right)-j\sum_{n=0}^{N-1}X(k) \cdots cos \left(\frac{2\pi kn}{N} \right)\]
where as its dual , IFFT is given by
The equation for FFT and IFFT differ by the co-efficients they take and the minus sign. Both equation does the same thing. They multiply the incoming signal with a series of sinusoids and separates them into bins.In fact, FFT and IFFT are dual and behaves in a similar way.IFFT and FFT blocks are interchangeable.
Since the OFDM signal ( c(t) in the equation above ) is in time domain, IFFT is the appropriate choice to use in the transmitter, which can be thought of as converting frequency domain samples to time domain samples. Well, you might ask : s(t) is not in frequency domain and they are already in time domain; so whats the need to convert it into time domain again ? The answer is IFFT/FFT equation comes handy in implementing the conversion process and we can eliminate the individual sinusoidal multipliers required in the transmitter/receiver side. The following figure illustrates, how the use of IFFT in the transmitter eliminates the need for separate sinusoidal converters. Always remember that IFFT and FFT blocks in the transmitter are interchangeable as long as their duals are used in receiver.
OFDM implementation using FFT and IFFT
The entire architecture of a basic OFDM system with both transmitter and receiver will look like this
A Complete OFDM communication system
An OFDM system is defined by IFFT/FFT length – N ,the underlying modulation technique ( BPSK/QPSK/QAM), supported data rate, etc..,. The FFT/IFFT length N defines the number of total subcarriers present in the OFDM system. For example, an OFDM system with N=64 provides 64 subcarriers. In reality, not all the subcarriers are utilized for data transmission. Some subcarriers are reserved for pilot carriers (used for channel estimation/equalization and to combat magnitude and phase errors in the receiver) and some are left unused to act as guard band. OFDM system do not transmitany data on the subcarriers that are near the two ends of the transmission band ( Not necessarily at the ends of the bands, implementation may differ). These subcarriers are collectively called guard band. The reservation of subcarriers to guard bands helps to reduce the out of band radiation and thus eases the requirements on transmitter front-end filters.The subcarriers in the guard band are also called Null subcarriers or virtual subcarriers.
For example IEEE 802.11 standard[1]specifies the following parameters for its OFDM physical layer.
FFT/IFFT size = 64 ( implies 64 subcarriers in total = used + unused = N_{fft})
Number of data subcarriers = 48 (\( N_d\))
Number of pilot subcarriers = 4 (\(N_p\))
Derived parameters from the above specification.
Number of total USED subcarriers = 52 (\( N_u = N_d+ N_p \))
Number of total UNUSED subcarriers = 12 (\( N_{un} = N_{fft} – N_u\)).
According to the spec, these 52 used subcarriers are distributed in the following way. The 52 used subcarriers are named as 1,2,3,…,26 and -1,-2,-3,…,-26. The used subcarriers 1 to 26 are mapped to 1 to 26 of IFFT inputs and the subcarriers -1,-2,..,-26 are mapped to the IFFT inputs 38 to 63. The remaining IFFT inputs 27 to 37 and the input 0 (dc input) are set to 0 .In this manner the 12 null subcarriers are mapped to IFFT inputs.
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In modulations, information is mapped on to changes in frequency, phase or amplitude (or a combination of them) of a carrier signal. Multiplexing deals with allocation/accommodation of users in a given bandwidth (i.e. it deals with allocation of available resource). OFDM is a combination of modulation and multiplexing. In this technique, the given resource (bandwidth) is shared among individual modulated data sources. Normal modulation techniques (like AM, PM, FM, BPSK, QPSK, etc.., ) are single carrier modulation techniques, in which the incoming information is modulated over a single carrier. OFDM is a multicarrier modulation technique, which employs several carriers, within the allocated bandwidth, to convey the information from source to destination. Each carrier may employ one of the several available digital modulation techniques (BPSK, QPSK, QAM etc..,).
Why OFDM
OFDM is very effective for communication over channels with frequency selective fading ( different frequency components of the signal experience different fading). It is very difficult to handle frequency selective fading in the receiver , in which case, the design of the receiver is hugely complex. Instead of trying to mitigate frequency selective fading as a whole (which occurs when a huge bandwidth is allocated for the data transmission over a frequency selective fading channel), OFDM mitigates the problem by converting the entire frequency selective fading channel into small flat fading channels (as seen by the individual subcarriers). Flat fading is easier to combat (compared to frequency selective fading) by employing simple error correction and equalization schemes.
Difference between FDM and OFDM:
OFDM is a special case of FDM ( Frequency Division Multiplexing). In FDM, the given bandwidth is subdivided among a set of carriers. There is no relationship between the carrier frequencies in FDM. For example, consider that the given bandwidth has to be divided among 5 carriers (say a,b,c,d,e). There is no relationship between the subcarriers ; a,b,c,d and e can anything within the given bandwidth.
If the carriers are harmonics, say (b=2a,c=3a,d=4a,d=5a , integral multiple of fundamental component a ) then they become orthogonal. This is a special case of FDM, which is called OFDM (as implied by the word – ‘orthogonal’ in OFDM)
Designing OFDM Transmitter:
Consider that we want to send the following data bits using OFDM : D = {d0,d1,d2,…). The first thing that should be considered in designing the OFDM transmitter is the number of subcarriers required to send the given data. As a generic case, lets assume that we have N subcarriers. Each subcarriers are centered at frequencies that are orthogonal to each other (usually multiples of frequencies).
The second design parameter could be the modulation format that we wish to use. An OFDM signal can be constructed using anyone of the following digital modulation techniques namely BPSK, QPSK, QAM etc.., The data (D) has to be first converted from serial stream into parallel stream depending on the number of sub-carriers (N). Since we assumed that there are N subcarriers allowed for the OFDM transmission, we name the subcarriers from 0 to N-1. Now, the Serial to Parallel converter takes the serial stream of input bits and outputs N parallel streams (indexed from 0 to N-1). These parallel streams are individually converted into the required digital modulation format (BPSK, QPSK, QAM etc..,). Lets call this output S0,S1,..SN. The conversion of parallel data (D) into the digitally modulated data (S) is usually achieved by a constellation mapper, which is essentially a look up table (LUT). Once the data bits are converted to required modulation format, they need to be superimposed on the required orthogonal subcarriers for transmission. This is achieved by a series of N parallel sinusoidal oscillators tuned to N orthogonal frequencies (f0,f1,…fN-1). Finally, the resultant output from the N parallel arms are summed up together to produce the OFDM signal.
The following figure illustrates the basic concept of OFDM transmission (note: In order to give a simple explanation to illustrate the underlying concept,the usual IFFT/FFT blocks that are used in actual OFDM system, are not used in the block diagram) .
OFDM Transmitter
Example:
The first example illustrates the concept of OFDM transmission with BPSK modulation as its underlying modulation format. The second example illustrates the OFDM transmission with pi/4 shifted QPSK modulation. Here 5 orthogonal subcarriers are assumed for the OFDM transmission.
The phenomenon of Rayleigh Flat fading and its simulation using Clarke’s model and Young’s model were discussed in the previous posts. The performance (Eb/N0 Vs BER) of BPSK modulation (with coherent detection) over Rayleigh Fading channel and its comparison over AWGN channel is discussed in this post.
We first investigate the non-coherent detection of BPSK over Rayleigh Fading channel and then we move on to the coherent detection. For both the cases, we consider a simple flat fading Rayleigh channel (modeled as a – single tap filter – with complex impulse response – h). The channel also adds AWGN noise to the signal samples after it suffers from Rayleigh Fading.
The received signal y can be represented as
$$ y=hx+n $$
where n is the noise contributed by AWGN which is Gaussian distributed with zero mean and unit variance and h is the Rayleigh Fading response with zero mean and unit variance. (For a simple AWGN channel without Rayleigh Fading the received signal is represented as y=x+n).
Non-Coherent Detection:
In non-coherent detection, prior knowledge of the channel impulse response (“h” in this case) is not known at the receiver. Consider the BPSK signaling scheme with ‘x=+/- a’ being transmitted over such a channel as described above. This signaling scheme fails completely (in non coherent detection scheme), even in the absence of noise, since the phase of the received signal y is uniformly distributed between 0 and 2pi regardless of whether x[m]=+a or x[m]=-a is transmitted. So the non coherent detection of the BPSK signaling is not a suitable method of detection especially in a Fading environment.
Coherent Detection:
In coherent detection, the receiver has sufficient knowledge about the channel impulse response.Techniques like pilot transmissions are used to estimate the channel impulse response at the receiver, before the actual data transmission could begin. Lets consider that the channel impulse response estimate at receiver is known and is perfect & accurate.The transmitted symbols (‘x’) can be obtained from the received signal (‘y’) by the process of equalization as given below.
$$ \hat{y}=\frac{y}{h}=\frac{hx+n}{h}=x+z $$
here z is still an AWGN noise except for the scaling factor 1/h. Now the detection of x can be performed in a manner similar to the detection in AWGN channels.
The input binary bits to the BPSK modulation system are detected as
where h(τ,t) is the time varying impulse response of the multipath fading channel having L multipaths and hi(t) and τi(t) denote the time varying complex gain and excess delay of the i-th path. The above mentioned impulse response can be implemented as a FIR filter as shown below :
Multipath Fading phenomena – modelled as a Time Varying FIR Filter
The channel under consideration can be modeled as a multipath fading channel in which the impulse response may follow distributions like Rayleigh distribution ( in which there is no Line of Sight (LOS) ray between transmitter and receiver) or as Rician distribution ( dominant LOS path exist between transmitter and receiver), Nagami distribution, Weibull distribution etc.
Different methods of simulation techniques were proposed to simulate/model multipath channels. Some of the models include clarke’s reference model, Jake’s model, Young’s model , filtered gaussian noise model etc.
A Rayleigh fading channel (flat fading channel) is considered in this text.For simplicity we fix the excess delays τi(t) in the above equation and we generate hi(t) that follows Rayleigh distribution. In this simulation Clarke’s Rayleigh fading model is used. This model is also called mathematical reference model and is commonly considered as a computationally inefficient model compared to Jake’s Rayleigh Fading simulator.
Theory of Rayleigh Fading:
Lets denote the complex impulse response h(t) of the flat fading channel as follows :
$$ h(t) = h_{I}(t) + jh_{Q}(t) $$
where hI(t) and hQ(t) are zero mean gaussian distributed. Therefore the fading envelope is Rayleigh distributed and is given by
We will use the Clarke’s Rayleigh Fading model (given below) and check the statistical properties of the random process generated by the model against the statistical properties of Rayleigh distribution (given above).
Clarke’s Rayleigh Fading model:
The random process of flat Rayleigh fading with M multipaths can be simulated with the sum-of-sinusoid method described as
Simulation:
1) The rayleigh fading model is implemented as a function in matlab with following parameters:
M=number of multipaths in the fading channel, N = number of samples to generate, fd=maximum Doppler spread in Hz, Ts = sampling period.
function [h]=rayleighFading(M,N,fd,Ts)
% function to generate rayleigh Fading samples based on Clarke's model
% M = number of multipaths in the channel
% N = number of samples to generate
% fd = maximum Doppler frequency
% Ts = sampling period
% Author : Mathuranathan for https://www.gaussianwaves.com
%Code available in the ebook - Simulation of Digital Communication Systems using Matlab
2)The above mentioned function is used to generate Rayleigh Fading samples with the following values for the function arguments. M=15; N=10^5; fd=100 Hz;Ts=0.0001 second;
Investigation of Statistical Properties of samples generated using Clarke’s model:
3) Mean and Variance of the real and imaginary parts of generated samples are
Mean of real part ~=0
Mean of imag part ~=0
Variance of real part = 0.4989 ~=0.5
Variance of imag part = 0.4989 ~=0.5
The results implies that the mean of the real and imaginary parts are same and are equal to zero.The variance of the real and imaginary parts are approximately equal to 0.5.
4)Next, the pdf of the real part of the simulated samples are plotted and compared against the pdf of Gaussian distribution (with mean=0 and variance =0.5)
Real Part of simulated samples exhibiting Gaussian Distribution characteristics
5)The pdf of the generated Rayleigh fading samples are plotted and compared against pdf of Rayleigh distribution (with variance=1)
PDF of simulated Rayleigh Fading Samples
6) From 4) and 5) we confirm that the samples generated by Clarke’s model follows Rayleigh distribution (with variance = 1) and the real and imaginary part of the samples follow Gaussian distribution (with mean=0 and variance =0.5).
7) The Magnitude and Phase response of the generated Rayleigh Fading samples are plotted here.
The Magnitude and Phase response of the generated Rayleigh Fading samples
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![\begin{matrix} h_{I}(nT_{s})=\frac{1}{\sqrt{M}}\sum_{m=1}^{M}cos\left \{ 2\pi f_{D}\,cos\left [ \frac{(2m-1)\pi + \theta}{4M} \right ].nT_{s}+\alpha_{m} \right \}\\ h_{Q}(nT_{s})=\frac{1}{\sqrt{M}}\sum_{m=1}^{M}sin\left \{ 2\pi f_{D}\,cos\left [ \frac{(2m-1)\pi + \theta}{4M} \right ].nT_{s}+\beta_{m} \right \} \\ h(nT_{s}) = h_{I}(nT_{s})+jh_{Q}(nT_{s}) \\\\ where \; \theta , \; \alpha_{m} \; and \; \beta_{m}\; are\; uniformly \;distributed\; over\; [0,2\pi)\\ for \;all\; n \;and\; are\; mutually\; distributed , \; \\ f_{D} = maximum\; Doppler\; spread, \; \\ T_{s}=sampling\; period, \; n=sample \; index \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D+h_%7BI%7D%28nT_%7Bs%7D%29%3D%5Cfrac%7B1%7D%7B%5Csqrt%7BM%7D%7D%5Csum_%7Bm%3D1%7D%5E%7BM%7Dcos%5Cleft+%5C%7B+2%5Cpi+f_%7BD%7D%5C%2Ccos%5Cleft+%5B+%5Cfrac%7B%282m-1%29%5Cpi+%2B+%5Ctheta%7D%7B4M%7D+%5Cright+%5D.nT_%7Bs%7D%2B%5Calpha_%7Bm%7D+%5Cright+%5C%7D%5C%5C+h_%7BQ%7D%28nT_%7Bs%7D%29%3D%5Cfrac%7B1%7D%7B%5Csqrt%7BM%7D%7D%5Csum_%7Bm%3D1%7D%5E%7BM%7Dsin%5Cleft+%5C%7B+2%5Cpi+f_%7BD%7D%5C%2Ccos%5Cleft+%5B+%5Cfrac%7B%282m-1%29%5Cpi+%2B+%5Ctheta%7D%7B4M%7D+%5Cright+%5D.nT_%7Bs%7D%2B%5Cbeta_%7Bm%7D+%5Cright+%5C%7D+%5C%5C+h%28nT_%7Bs%7D%29+%3D+h_%7BI%7D%28nT_%7Bs%7D%29%2Bjh_%7BQ%7D%28nT_%7Bs%7D%29+%5C%5C%5C%5C+where+%5C%3B+%5Ctheta+%2C+%5C%3B+%5Calpha_%7Bm%7D+%5C%3B+and+%5C%3B+%5Cbeta_%7Bm%7D%5C%3B+are%5C%3B+uniformly+%5C%3Bdistributed%5C%3B+over%5C%3B+%5B0%2C2%5Cpi%29%5C%5C+for+%5C%3Ball%5C%3B+n+%5C%3Band%5C%3B+are%5C%3B+mutually%5C%3B+distributed+%2C+%5C%3B+%5C%5C+f_%7BD%7D+%3D+maximum%5C%3B+Doppler%5C%3B+spread%2C+%5C%3B+%5C%5C+T_%7Bs%7D%3Dsampling%5C%3B+period%2C+%5C%3B+n%3Dsample+%5C%3B+index+%5Cend%7Bmatrix%7D+&bg=ffffff&fg=0000A0&s=2&c=20201002)
