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binary phase shift keying is one of a modulation technique used in communication technology and simplest form of PSK. It is a type of phase modulation using 2 distinct carrier phases to signal ones and zeros. It uses two phases which are separated by 180Â° and so can also be termed 2-PSK.when modem implementation, BPSK modulation technique also available.
Modem implementation is rapidly growing technology with modern communication technology. The one word 'MODEM' stands for 'modulator-demodulator'. Many modulation techniques are available for modem design. It is Depending on the system requirement and equipment.
OFDM is the most appropriate transmission method for future generation wireless communications.
So, implement this modem by using mathlab simulink. All above components are available from simulink software.
Chapter 2 looks into the Theoretical review and principles of Wireless communication, BPSK modulation, Convolutional coding and decoding, OFDM scheme and wireless LAN Specification.
Chapter 3 discussed about Simulink implementation part. And also have disscuss in this chapter Matlab/simulink Overview, about setting up block parameters, AWGN, FFT, Convolutional encoder Viterbi decoder and all about model blocks and its specifications. And also have discussed about Model blocks specifications and parameters, About the implemented system model BPSK, OFDM Transceiver.
Chapter 4 discussed about Testing and result. Scatter plot diagrams spectrum scope diagrams, scope result, BER results, BER Curve and all about system performance had discuss using this chapter.
Chapter 5 discussed about overall project and further development scheme.
Chapter 2- Literature Review
2.1 Wireless communication
There are many sections in communication technology which are dramatically growing. Among them Wireless communications Technology is considerable. This provides high speed and high quality information transform between movable devices located anywhere in the earth.
Multimedia Internet-enabled cell phones, smart phones, automated highway systems, video teleconferencing and distance learning, and autonomous sensor networks are some of few potential applications of this industry.
Yet Wireless techniques include a remarkable technical challenge for supporting these applications.
This course will cover advanced topics in wireless communications for voice, data, and multimedia.
First of all short overview of current wireless systems and standards. Then we have to characterize the wireless channel, including path loss for different environments, random log-normal shadowing due to signal attenuation, and the flat and frequency-selective properties of multipath fading.
Next we analyze the basic capacity limits of wireless channels and the characteristics of the capacity-achieving transmission strategies. Moreover typically these strategies aren't practical. Therefore, our next focus will be on practical digital modulation techniques and their performance under wireless channel impairments, including flat and frequency selective fading.
Analyze techniques to improve the speed and performance of wireless technologies is the next part.
We will examine the design and performance of reconciling modulation and diversity techniques to compensate for flat-fading. Three techniques to combat frequency-selective fading are then investigates: adaptive equalization, multicarrier modulation, and spread spectrum. We will also study the multiple access capabilities of spread spectrum with multiuser detection. The course finishes with a short overview of wireless networks, considering multiple and random access techniques, WLANs, cellular system design, and ad-hoc network design. The other things which we going to discussed are applications for those systems, including the evolution of cell phones and PDAs, smart phones and appliances, sensor networks, and automated highways and skyways. 
2.2 Modulation and demodulation
Modulation is the modifying of a signal to carry intelligent data over the communications channel. Several types of modulation Techniques are available, depending on the system requirement and equipment. Â The most commonly used types of modulation techniques are amplitude modulation, frequency modulation, and Â Â phase modulation. Demodulation is the act of returning modulated data signals to their original form. 
Amplitude modulation refers to modifying the amplitude of a sine wave to store data.
Frequency modulation Â refers Â to Â changing Â the Â frequency Â of Â a signal Â to Â indicate Â a Â logic Â 1 Â or Â a Â logic Â 0. One frequency indicates logic 1, and the other frequency indicates logic 0.
Â Phase modulation is more Â Â complex Â Â than Â Â frequency modulation(FM) or amplitude Â modulation(AM).Phase Â modulation Â uses Â a signal Â frequency Â sine Â wave Â and Â performs Â phase shifts of Â the Â sine Â wave Â to Â store Â data. Â A modification of phase modulation involves the use of several discrete phase shifts to indicate the state of two or more data bits. 
A modem is supposes to convert digital information to analog signals. And to convert analog signals back in to useful digital information. It mean modem stand for modulator-Demodulator. It is a device that modulates and demodulates data in a digital communication system. Modems are available in a variety of types, with various speeds and capabilities. A modem consists of two functionally separate areas. The transmitter section prepares, or modulates the data for transmission. The receiver section demodulators, or returns, incoming data to its original form.
2.4 Binary Phase Shift Keying (BPSK)
Binary Phase Shift Keying (B-PSK) is the simplest form of PSK. It is represented by two different phases typically 0. BPSK is the one of most strong type of Phase Shift Keying and its takes very high levels of distortion for the demodulator to reach a wrong decision. However BPSK only able to transmit 1bit/symbol therefore it is not suitable of high data rate applications. It does not particularly stuff where the constellation points are located.
2.5 Convolution coding
Convolutional coding technique is one of a special case of error-control coding. A convolutional encoder is not a memoryless device. Even though a convolutional coder accepts a fixed number of message symbols and produces a fixed number of code symbols, its computations depend not only on the current set of input symbols but on some of the previous input symbols.
Convolution Coding Block: G1:171(octal) for And G2:133(octal) for B. The generator sequences G1 and G2 can be derived as below
Paths which are selected for binary summation are elected by '1' and those which are not selected are designated by '0'. Moving from right to left, for A output, the generator sequence will be 1001111. Appending 2 '0' to the right, we get, 1101101. Reading this from right to left, G1 = 171(octal). Similarly, we can derive G2 = 133(octal).
The default rate of Convolution Encoding is Â½, since for a given input; we get 2 outputs, A and B.
Figure 2-1: Convolutional encoder (k = 7)
2.6 Orthogonal Frequency Division Multiplexing (OFDM)
The OFDM system came to light as a result of investigate into combating the effects of ISI. The basic principles in the design of the OFDM were the same as FDM. FDM technique uses multiple frequencies to transmit multiple signals in parallel. Each signal has its own frequency range (subcarriers) which is then modulated by the data. The subcarriers are then separated by a guard period to ensure that they do not overlap. These subcarriers are then separated at the receiver using filters to separate the bands. 
OFDM is very similar to FDM except that it is spectrally more efficient. It places the subcarriers so closely together that they actually overlap each other, yet do not interfere. This is done by finding frequencies which are orthogonal to each other, meaning that they do not spectrally overlap.
In OFDM each symbol contains subcarriers that are nonzero over a T-second interval. Each subcarrier has exactly an integer number of cycles in this time interval and the number of cycles between adjacent subcarriers differs by one. This property is true from the time domain perspective. From the frequency domain perspective, each subcarrier has each maximum spectrum value at its centre frequency and zero at the centre frequency of other subcarriers. Since the OFDM receiver calculates the spectrum values at those points
That correspond to the maxima of individual subcarriers, it can demodulate each subcarrier free from any interference from the other subcarriers
Figure 2-2: example of OFDM Spectrum of 5 subcarriers
The above figure shows the spectrum of a single OFDM subcarrier and the spectrum of 5 subcarriers. It is clearly seen that the subcarriers are overlapping each other, yet in real time transmission this overlapping has no negative effects due to the orthogonality technique.
The most important application of OFDM remains in wireless LAN communications. Starting with the IEEE 802.11 WLAN, the IEEE802.16 and the ETSI Broad Radio Access Networks (BRAN) also use OFDM. In Japan research is being undertaken to provide ultra-high-speed wireless indoor LANs. 
2.7 Wireless LAN Specification- IEEE 802.11 Standard
11 September 2009 --IEEE said that its Standards Board has ratified the IEEE 802.11n -2009 modification, defining mechanisms that present significantly improved data rates and ranges for wireless local area networks (WLANs). This new amendment to the IEEE 802.11 base standard is designed to help the data communications industry address the increasing demands placed on enterprise, public or home WLANs with the increase of higher-bandwidth file transfers and next-generation multimedia applications. IEEE 802.11 based WLANs are widely deployed, with more than 1 million units shipping per day. 
The IEEE 802.11 standard defines how to design interoperable WLAN equipment that provides a variety of capabilities including a wide range of data rates, quality of service, reliability, range optimization, device link options, network management and security.
 relevant project
Chapter 3 -Simulink Implementation
3.1 Simulink Overview
Simulink is most suitable for multidomain simulation and Model-Based Design environment. As well as simulink works dynamic and fixed systems. Simulink gives interactive graphical settings. Customer can use set of block libraries and try to do his design, simulate, implement, and test a range of time-shifting systems. We can use simulink for communications implementation, system controls, signal processing, video processing, and image processing. 
3.2 Setting up of block parameters
The simulink structure block diagram is shown in following Figure xxx. In order to do system level simulation, a transmitter-channel-receiver chain is modelled in using Matlab/Simulink software. These following blocks are explained below step by step.
End to end BPSK Modem Communication System.
All wireless transceiver settings the simulation using by a data generator, a transmitter, a wireless communication channel and a receiver. As the performance of the system I have tested a Bit Error Rate using (BER) calculator was also added to the design.
The IEEE 802.11 SIGNAL field, this BPSK-OFDM system have use binary Bernoulli generator as a system bit generator with a convolution encoder of coding rate R = 1/2, matching to the desired data rate.
Major parameters of the OFDM
Information data rate
6, 9, 12, 18, 24, 36, 48,
and 54 Mb/s
(6, 12, and 24 Mb/s are
(20 MHz channel
3, 4.5, 6, 9, 12, 18, 24,
and 27 Mb/s
(3, 6, and 12 Mb/s are
(10 MHz channel
1.5, 2.25, 3, 4.5, 6, 9, 12,
and 13.5 Mb/s
(1.5, 3, and 6 Mb/s are
(5 MHz channel
Error correcting code
K = 7 (64 states)
K = 7 (64 states)
K = 7 (64 states)
1/2, 2/3, 3/4
1/2, 2/3, 3/4
1/2, 2/3, 3/4
Number of subcarriers
Major parameters of OFDM 
For this system have use following specific IEEE 802.11 parameters.
OFDM System parameters
Number of FFT points
Number of sub-carriers
Number of data sub-carriers
Number of pilot sub-carriers
Â½ convolutional, constraint length 7, optional puncturing
6, 9, 12, 18, 24, 36, 48, 54 Mbps
The system uses 64 point FFT. The OFDM frame duration have use 80 chips. 64 is for data with16 cyclic prefix. Out of the 64 narrow-band sub-carriers, only 52 are carrying signal and other 12 are zeros. Four of the 52 sub-carriers are used as pilots and the other 48 are used for data. finaly system have use 48 data,4 pilot sub carrier,12 guard band and 16 cyclic prefix. Total OFDM receiver output was 80.Using different modulation scheme mixed with puncturing of the convolutional encoder, variable data rate can be achieved with a minimum of 6 Mbps and maximum of 54 Mbps.
4.3 BPSK modulation
The system used following blocks from the Matlab Simulink Library Browser into the model window, and put in these blocks into the model. Binary Phase Shift Keying (BPSK) Modulator Baseband block from Digital Baseband Modulation sub library. AWGN Channel block from the Channels library. BPSK Demodulator block from Digital Baseband Modulation sublibrary of the Modulation library. Bernoulli binary generator block, from Random Data Sources sublibrary of Communication Sources. Error rate calculation block, from communication sinks library. And also user can find display block from sink library and Scope block from sinks Library.
When connect these blocks final implemented model as follows.
BPSK communication system model
Figure 3-1: BPSK communication system model
Bernoulli Binary Generator
The Bernoulli Binary Generator block generates random binary numbers.When brows the Bernoulli Binary Generator block we can see as follows.
Figure 3-2: bernoulli binary generator dialog box
Probability of a zero- zero output probability.
Sample time- The period of each row of a frame based matrix or sample based vector.
Frame-based outputs-Determines whether the output is frame-based or sample-based.
Samples per frame -The number of samples per each frame based signal.
Output data type - Bernoulli binary generator block available these data types output form as a int16, boolean, int32, int8, uint8, uint16, uint32, single, or double. Default block set is double
Binary Phase Shift Keying (Bpsk modulator baseband)
Figure 3-3: BPSK Modulator baseband block
The above BPSK Modulator and Demodulator Baseband blocks designed using binary phase shift keying (BPSK) modulation. BPSK is a process for modulating a binary signal onto a complex waveform by shifting the phase of the complex signal.
User can set the value of in the Phase offset limitation using dialog boxes for these BPSK Modulator Baseband block and the BPSK Demodulator Baseband block.
Supported data type (Output)
The BPSK Modulator output data type can be set to double, single, fixed point, User defined, or Inherit via back propagation.
Supported Data Types (input)
The BPSK Modulator output data type can be set to Double Single-precision floating point, Boolean 8-, 16-, and 32-bit signed integers,/8-, 16-, and 32-bit unsigned integers precision floating point.
BPSK Demodulator baseband
Figure 3-4: BPSK Demodulator baseband block
The above BPSK Demodulator Baseband blocks designed using binary phase shift keying (BPSK) demodulation. The input is a baseband representation of the modulated signal.
We can change decision type as hard decision and soft decision.
When use Decision type is set to hard decision, the output data type can be set to inherit via internal rule, Smallest unsigned integer, Boolean, double, single, int8, int16, uint16, int32, uint8 or uint32.
Default settings parameter is Inherit via internal rule. When user chooses default settings the block will inherit the output data type from the input port. In this situation output data type will be the same as the input data type if the input is a floating point type single or double. If the input data type is fixed point, the output data type will work as if this parameter is set to smallest unsigned integer.
Supported Data Types
Input- The BPSK Demodulator input data type can be set to Double-precision floating point, Single-precision floating point, Signed Fixed point (this is only for Hard decision mode)
Output- The BPSK Demodulator output data type can be set to Double precision floating point, Single precision floating point Boolean, 8-, 16-, and 32 bit signed integers 8-, 16-, and 32 bit unsigned integers
Channel(while Gaussian noise)
Figure 3-5: BPSK Modulator baseband block
When signal pass through communication system add some unwanted signal. It can be introduce as noise, interference, fading and distortions. This situation affects every real communication system. For match real world communication situation and simulation situation user can use channel. This implementation used adds while Gaussian noise (AWGN) as a channel. This channel adds while Gaussian noise to signal with the specified value of Eb/No. Typical. The value of SNR is inversely proportional to the BER.
BER Calculator- a bit error calculator was added to measure the number of BER vs. Eb/No for the various simulations to be carried out.
4.4 Convolution encoding and viterbi decoding
When added convolution encoder and viterbi decoder as an Error correction blocks for the implemented above bpsk model, new appear as follows.
Figure 3-6: BPSK communication system model with Convolutional coding
Convolutional encoder and viterbi decoder have use as error correction method, for this implementation. Convolutional coding method is a particular case of error manages Coding.
Even though convolution coder accepts a fixed number of message symbols and Produces a fixed number of code symbols, it calculation depend not only on some of the previous input symbols but also on the existing situate of Input symbols. Convolutional encoder uses poly2 trellis function to produce a trellis by using the code generator, constraint length and feedback connection. In this model the coding rate is Â½ and the constraint length is 7. This block can process multiple symbols at a time. 
Figure 3-7: Convolutional encoder block
Input and Output Sizes
We can use same sizes for input and output of this block. The convolutional encoder block supports double, single, int8, boolean, int16, uint16, int32, uint8, uint32, and ufix1. The port data types are inherited from the signals that drive the block. The input reset port supports double and boolean typed signals. 
Specifying the Encoder
User can define the convolutional encoder, using the Trellis structure limitations. If user wants to specify the encoder using its generator polynomials, constraint length, and possibly feedback connection polynomials, uses a poly2trellis command within the Trellis structure field. For example, to use an encoder with a constraint length of 7, code generator polynomials of 171 and 133 (in octal numbers), and a feedback connection of 171 (in octal), set the Trellis structure parameter to poly2trellis (7, [171 133], 171) 
Convolutional encoder dialog box
Figure 3-8: Convolutional encoder dialog box
Trellis structure - MATLAB structure that contains the trellis description of the convolutional encoder.
Output final state -When user select Output final condition, the second output port signal specifies the output state for the block. The output signal is a scalar, integer value. User can select Output final state for all operation modes except Terminate trellis by appending bits.
Puncture code - Selecting this option opens the field Puncture vector.
Puncture vector- Vector used to puncture the encoded data. The puncture vector is a pattern of 1s and 0s where the 0s specify the punctured bits. This field shows when select Punctured code.
Figure 3-9: Viterbi Decoder block
The Viterbi Decoder block decodes input symbols to produce binary output symbols. This block can process several symbols at a time for faster performance. viterbi decoder dialog box, shown as follows.
Figure 3-10: Viterbi Decoder Dialog Box
Trellis structure-MATLAB arrangement that contains the trellis description of the convolutional encoder. Using parameters in here and should match with Convolutional Encoder block parameters.
Punctured code -Select this check box to specify a punctured input code.
Puncture vector -stable puncture model vector applied at the transmitter. The puncture vector is a pattern of 1s and 0s. 0s shows the punctured bits. This field shows when the check box Punctured code is selected.
Decision type - Unquantized, Hard Decision, or Soft Decision.
Number of soft decision bits -The number of soft decision bits used to represent each input. This field is active only when Decision type is set to Soft Decision.
Traceback depth - The number of trellis branches used to construct each traceback path.
Operation mode- Method for transitioning between successive input frames. For frame-based input, the choices are Continuous, Terminated, and Truncated. Sample-based input must use the Continuous mode.
Enable reset input port -When you check this box, the decoder opens an input port labeled Rst. providing a nonzero input value to this port causes the block to set its internal memory to the initial state before processing the input data.
Output data type
The output signal's data type can be double, single, boolean, int8, uint8, int16, uint16, int32, uint32, or set to Inherit via internal rule or Smallest unsigned integer.
Overview of the Simulations
The above two simulations have a related formation. A data source generates a random binary sequence that is convolutionally encoded, BPSK modulated, and passed through an AWGN channel. After the decoding and the simulation compares the received decoded symbols with the original transmitted symbols in order to compute the bit error rate.
3 .6 BPSK modulation convolution coding with delay unit
When signal go through the convolutional encoder, output signal shows delay. To solve this Problem we can use Delay block.
BPSK Communication system (convolutional coding, delay unit)
Figure 3-11: BPSK Communication system (convolutional coding, delay unit)
4.5 BPSK modulation convolution coding with OFDM
This is final BPSK communication system using matlab/simulink. This system contains suitable error correction method, relevant multiplexing method and error rate calculation. And also have observed scatter plot diagram, spectrum scope diagram and scope diagram. This simulated system worked properly.
Communication System Implemented with Orthogonal Division Multiplexing OFDM
Figure 3-12: Communication System Implemented with Orthogonal Division Multiplexing (OFDM).
Orthogonal Division Multiplexing (OFDM) Transmitter
This system consists of remove cyclic Prifix, FFT, Frame conversion, select rows block and the Cyclic prefix addition block. The system can represent OFDM transmission uses 6 sub-carriers, 4 pilot- carriers, 12 Guard and and a 16- sample cyclic prefix. OFDM transmitter sub system is describe in Figure 3-13
Figure 3-13: Orthogonal Division Multiplexing (OFDM) Transmitter
Set into the corresponding time domain representation of the data. It is the basic idea of the multi-carrier modulation. The function of the zero
The number of pilots used in this OFDM system depends on the characteristics of the channel through which the signal is sent. For 64 subcarriers case in 802.11a the 4 pilot carrier positions is -21,-7, 7, and 21. Pilot sub-carriers used to prevent frequency and phase shift errors. Guard band used for remove channel interference.
The number of pilot carriers and zero carriers (Guard bands) depends on the modulation scheme you are using for a particular stream of data and channel conditions. This model has use 12 zero carriers.
IFFT block serves inverse Fourier transform to convert the frequency domain data. Cyclic prefix block adds to avoid mixing of subsequent symbols in the receiver. The cyclic prefix can ensure that delay
Model of the OFDM symbol always have an integer number of cycles with the FFT interval. As cyclic prefix, the last 6 sub-carriers are copied into the beginning of the OFDM symbols by using selector block
Multiport selector Dialog box
Figure 3-14: Multiport Selector Dialog box
Above Figure shows how to use selector Rows Block and how to put into the Dialog Box
Guard bands are commonly used in Frequency Division Multiplexing (FDM) scheme. But it available for used in any data transmission method that relies on frequencies. A narrow frequency band between adjacent channels in multiplexing that is kept unused to prevent the channels from overlapping and causing crosstalk among modulated signals. 
An OFDM system treats the symbols at the transmitter end as if they are in frequency domain. These symbols are used as inputs to the IFFT block which converts the signal in time domain for further processing and transmission. 
The IFFT output is basically the summary of all the N orthogonal sinusoids input to it. Thus, the IFFT block provides a simple way to modulate data onto N orthogonal subcarriers. The block of N output samples from the IFFT makes up a single OFDM symbol..
Add cyclic prefix
Cyclic prefix block used to avoid intersymbol interference (ISI) which is a common .Signal when go through this block add further 16 carriers.
OFDM Receiver consists of remove cyclic prefix block, FFT block, Frame conversion block, remove zero block (selector) Selector Rows block and pilot carrier remove block performs the reverse functions of the OFDM transmitter.
Figure 3-15: Othoganal Frequency Division Multiplexing (OFDM) Receiver Model
Remove cyclic prefix block removes cyclic prefix added in the transmission side. FFT block transforms the time domain data into frequency domain. Selector block removes the zero and then frame conversion block change to proper BPSK symbol at the output.
Matlab/Simulink FFT block available at communication transform block set Library.
Figure 3-16: FFT block
Figure 3-16: FFT dialog box
Output in bit-reversed order -Designate the order of the output channel elements relative to the ordering of the input elements.
Divide butterfly outputs by two -When select this parameter, the output of each butterfly of the FFT is divided by two. When you do not select this parameter, the block does not scale the output.
Inherit FFT length from input dimensions
Select to inherit the FFT length from the input dimensions. When you select this parameter, the input length P must be a power of two. When you do not select this parameter, the FFT length parameter becomes available.
Supported Data Types
Input - Double-precision floating point, Single-precision floating point, Fixed point, 8-, 16-, and 32-bit signed integers, 8-, 16-, and 32-bit unsigned integers
Output - Double-precision floating point ,Single-precision floating point, Fixed point (signed only),8-, 16-, and 32-bit signed integers
Multi port selector (Remove Pilots)
Using Multiport Selector user can remove pilot carriers and filter the sub carriers
X- Sub Carriers position
Y- Pilot Carriers Position
Figure 3-16 Multi port selector Dialog Box
When consider above figure, it have mentioned about Sub carrier and pilot carriers.
'1:5'- mentioned that 1st row.6th one is pilot position. Again '7:19' mentioned that 2nd Row. Again 6th position is pilot carrier position. According this pattern user can create this multiport selector.
4.7 Chapter summary
 The MathWorks, Inc.( 1984-2010) Modeling a Channel with Modulation [online]
Available at : http://www.mathworks.com/access/helpdesk/help/toolbox/commblks/gs/fp29851.html
 IEEE Std 802.11â„¢-2007 (Revision of IEEE Std 802.11-1999)
Chapter-5 testing and results
5.1 BPSK modulation testing and results
Bernoulli binary generator was used to generate the digital communication signal. It was connected to modulation block. Discrete-time scatter plot scope and scope were connected as Figure xxx
Figure 4-1 :Constellation before AWGN & after AWGN (when SNR=20)
The model displays a scatter plot of a signal with added noise.
Figure 4-2: BPSK Modulation cope result (when SNR=125)
5.2 BPSK modulation with convolution coding
Figure 4-3: BPSK modulation with convolution coding scope result(when SNR 125)
5.3 BPSK modulation, convolution coding with delay unit
Figure 4-4: BPSK modulation, convolution coding with delay unit
5.5 BPSK modulation, convolution coding plus OFDM
Figure 4-5: BPSK modulation, convolution coding plus OFDM. spectrum scope result before after IFFT Block
5.7 Chapter summary
Chapter-5 Conclusion and Further Development
BPSK is one of suitable and most commonly data transmission technology for wireless communications. This project successfully designed, implemented and simulated a BPSK wireless transmitter and the receiver .and also calculated its performance as required in the project aims and objectives. OFDM wireless transceiver Act main character in this implemented model. The Model performance of the BPSK and OFDM transceiver was measured according to the Bit Error Rate (BER) obtained in every simulation.
The Project First Investigate the Performance of the basic BPSK Communication system and obtained Scatter plot Result and scope Result. It was tested under AWGN channel. According to the scatter plot result and scope result gained idea BPSK modulated signal behaviour. After that, added step by step additional blocks to the model (Convolutionlan encoder, viterbi decoder, OFDM Transmitter and receiver sub systems) and obtained system performance results using scatter plot scope and spectrum scope. After that measured system performance according to the Bit Error Rate (BER). Hereafter using final model system performance BER values, measured system performance and obtained BER curve. Finally project simulink model BER Performance curve compared with theoretical performance curve.