Digital Modulations Archives - Page 2 of 2

QAM modulation: simulate in Matlab & Python

A generic complex baseband simulation technique, to simulate all M-ary QAM modulation techniques is given here. The given simulation code is very generic, and it plots both simulated and theoretical symbol error rates for all M-QAM modulation techniques.

Rectangular QAM from PAM constellation

There exist other constellation shapes (like circular, triangular constellations) that are more efficient (in terms of energy required to achieve same the error probability) than the standard rectangular constellation. Rectangular (symmetric or square) constellations are the preferred choice of implementation due to its simplicity in implementing modulation and demodulation.

In one of the earlier articles, I have discussed the method of constructing constellation for rectangular QAM modulation using Karnaugh-map walks, where the inherent property of Karnaugh-maps is exploited to construct Gray coded QAM symbols.

Any rectangular QAM constellation is equivalent to superimposing two Amplitude Shift Keying (ASK) signals (also called Pulse Amplitude Modulation – PAM) on quadrature carriers. For example, 16-QAM constellation points can be generated from two 4-PAM signals, similarly the 64-QAM constellation points can be generated from two 8-PAM signals.

*Figure 1: Signal space constellations for 16-QAM and 64-QAM*

*Figure 1: Signal space constellations for 16-QAM and 64-QAM*

The generic equation to generate PAM signals of dimension D is

For generating 16-QAM, the dimension D of PAM is set to . Thus for constructing a M-QAM constellation, the PAM dimension is set as . Matlab code for dynamically generating M-QAM constellation points based on Karnaugh map Gray code walk is given below. The resulting ideal constellations for Gray coded 16-QAM and 64-QAM are shown in Figure 1.

Matlab code

Full Matlab code available in the book Digital Modulations using Matlab – build simulation models from scratch

function [s,ref]=mqam_modulator(M,d)
%Function to MQAM modulate the vector of data symbols - d
%[s,ref]=mqam_modulator(M,d) modulates the symbols defined by the vector d
% using MQAM modulation, where M specifies order of M-QAM modulation and
% vector d contains symbols whose values range 1:M. The output s is modulated
% output and ref represents reference constellation that can be used in demod
if(((M˜=1) && ˜mod(floor(log2(M)),2))==0), %M not a even power of 2
  error('Only Square MQAM supported. M must be even power of 2');
end
  ref=constructQAM(M); %construct reference constellation
  s=ref(d); %map information symbols to modulated symbols
end

Python code

Full Matlab code available in the book Digital Modulations using Python

class QAMModem(Modem):
    # Derived class: QAMModem
    
    def __init__(self,M):
        
        if (M==1) or (np.mod(np.log2(M),2)!=0): # M not a even power of 2
            raise ValueError('Only square MQAM supported. M must be even power of 2')
        
        n = np.arange(0,M) # Sequential address from 0 to M-1 (1xM dimension)
        a = np.asarray([xˆ(x>>1) for x in n]) #convert linear addresses to Gray code
        D = np.sqrt(M).astype(int) #Dimension of K-Map - N x N matrix
        a = np.reshape(a,(D,D)) # NxN gray coded matrix
        oddRows=np.arange(start = 1, stop = D ,step=2) # identify alternate rows
        
        nGray=np.reshape(a,(M)) # reshape to 1xM - Gray code walk on KMap
        #Construction of ideal M-QAM constellation from sqrt(M)-PAM
        (x,y)=np.divmod(nGray,D) #element-wise quotient and remainder
        Ax=2*x+1-D # PAM Amplitudes 2d+1-D - real axis
        Ay=2*y+1-D # PAM Amplitudes 2d+1-D - imag axis
        constellation = Ax+1j*Ay
        Modem.__init__(self, M, constellation, name='QAM') #set the modem attributes

This article is part of the following books
● Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885
● Digital Modulations using Python ISBN: 978-1712321638
All books available in ebook (PDF) and Paperback formats

M-QAM demodulation (coherent detection)

Generally the two main categories of detection techniques, commonly applied for detecting the digitally modulated data are coherent detection and non-coherent detection.

In the vector simulation model for the coherent detection, the transmitter and receiver agree on the same
reference constellation for modulating and demodulating the information. The modulators generate the reference constellation for the selected modulation type. The same reference constellation should be used if coherent detection is selected as the method of demodulating the received data vector.

On the other hand, in the non-coherent detection, the receiver is oblivious to the reference constellation used at the transmitter. The receiver uses methods like envelope detection to demodulate the data.

The IQ detection technique is an example of coherent detection. In the IQ detection technique, the first step is to compute the pair-wise Euclidean distance between the given two vectors – reference array and the received symbols corrupted with noise. Each symbol in the received symbol vector (represented on a p-dimensional plane) should be compared with every symbol in the reference array. Next, the symbols, from the reference array, that provide the minimum Euclidean distance are returned.

Let x=(x₁,x₂,…,x_p) and y=(y₁,y₂,…,y_p) be two points in p-dimensional space. The Euclidean distance between them is given by

The pair-wise Euclidean distance between two sets of vectors, say x and y, on a p-dimensional space, can be computed using the vectorized code. The vectorized code returns the ideal signaling points from matrix y that provides the minimum Euclidean distance. Since the vectorized implementation is devoid of nested for-loops, the program executes significantly faster for larger input matrices. The given code is very generic in the sense that it can be easily reused to implement optimum coherent receivers for any N-dimensional digital modulation technique (Please refer the books Digital Modulations using Matlab and Digital Modulations using Python for complete simulation code) .

Matlab code

Full Matlab code available in the book Digital Modulations using Matlab

function [dCap]= mqam_detector(M,r)
%Function to detect MQAM modulated symbols
%[dCap]= mqam_detector(M,r) detects the received MQAM signal points
%points - 'r'. M is the modulation level of MQAM
   if(((M˜=1) && ˜mod(floor(log2(M)),2))==0), %M not a even power of 2
      error('Only Square MQAM supported. M must be even power of 2');
   end
   ref=constructQAM(M); %reference constellation for MQAM
   [˜,dCap]= iqOptDetector(r,ref); %IQ detection
end

Python code

Full Matlab code available in the book Digital Modulations using Python

Performance simulation results

The simulation results for error rate performance of M-QAM modulations over AWGN channel and Rician flat-fading channel is given in the following figures.

*Figure 2: Error rate performance of M-QAM modulations in AWGN channel*

*Figure 3: Error rate performance of 64-QAM modulation in Rician flat-fading channels having different values for Rician K-factors.*

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

Reference

[1] John G. Proakis, “Digital Communciations”, McGraw-Hill; 5th edition.↗

Books by the author

Wireless Communication Systems in Matlab Second Edition(PDF) (172 votes, average: 3.66 out of 5) Checkout Added to cart	Digital Modulations using Python (PDF ebook) (127 votes, average: 3.58 out of 5) Checkout Added to cart	Digital Modulations using Matlab (PDF ebook) (134 votes, average: 3.63 out of 5) Checkout Added to cart
Hand-picked Best books on Communication Engineering Best books on Signal Processing

MPSK modulation: simulate in Matlab & Python

A generic complex baseband simulation technique, to simulate all M-ary phase shift keying (M-PSK) modulation techniques is given here. The given simulation code is very generic, and it plots both simulated and theoretical symbol error rates for all MPSK modulation techniques.

M-ary phase shift keying (M-PSK) modulation

In phase shift keying, all the information gets encoded in the phase of the carrier signal. The M-PSK modulator transmits a series of information symbols drawn from the set m∈{1,2,…,M}. Each transmitted symbol holds k bits of information (k=log₂(M)). The information symbols are modulated using M-PSK mapping.

*Figure 1: Signal space constellations for various MPSK modulation*s

The general expression for a M-PSK signal set is given by

Here, M denotes the modulation order and it defines the number of constellation points in the reference constellation. The value of M depends on the parameter k – the number of bits we wish to squeeze in a single MPSK symbol. For example if we wish to squeeze in 3 bits (k=3) in one transmit symbol, then M = 2^k = 2³ = 8 and this results in 8-PSK configuration. M=2 gives binary phase shift keying (BPSK) configuration. The configuration with M=4 is referred as quadrature phase shift keying (QPSK). The parameter A is the amplitude scaling factor. Using trigonometric identity, equation (1) can be separated into cosine and sine basis functions as follows

This can be expressed as a combination of in-phase and quadrature phase components on an I-Q plane as

Normalizing the amplitude as , the points on the reference constellation will be placed on the unit circle. The MPSK modulator is constructed based on this equation and the ideal constellations for M=4,8 and 16 PSK modulations are shown in Figure 1.

Matlab code

Full Matlab code available in the book Digital Modulations using Matlab – build simulation models from scratch

function [s,ref]=mpsk_modulator(M,d)
%Function to MPSK modulate the vector of data symbols - d
%[s,ref]=mpsk_modulator(M,d) modulates the symbols defined by the
%vector d using MPSK modulation, where M specifies the order of
%M-PSK modulation and the vector d contains symbols whose values
%in the range 1:M. The output s is the modulated output and ref
%represents the reference constellation that can be used in demod
   ref_i= 1/sqrt(2)*cos(((1:1:M)-1)/M*2*pi);
   ref_q= 1/sqrt(2)*sin(((1:1:M)-1)/M*2*pi);
   ref = ref_i+1i*ref_q;
   s = ref(d); %M-PSK Mapping
end

Python code

Full Python code available in the book Digital Modulations using Python

modem.py: PSK modem - derived class
class PSKModem(Modem):
	# Derived class: PSKModem
	def __init__(self, M):
		#Generate reference constellation
		m = np.arange(0,M) #all information symbols m={0,1,...,M-1}
		I = 1/np.sqrt(2)*np.cos(m/M*2*np.pi)
		Q = 1/np.sqrt(2)*np.sin(m/M*2*np.pi)
		constellation = I + 1j*Q #reference constellation
		Modem.__init__(self, M, constellation, name='PSK') #set the modem attributes
.
.
.

M-PSK demodulation (coherent detection)

Generally the two main categories of detection techniques, commonly applied for detecting the digitally modulated data are coherent detection and non-coherent detection.

Let x=(x₁,x₂,…,x_p) and y=(y₁,y₂,…,y_p) be two points in p-dimensional space. The Euclidean distance between them is given by

Matlab code

Full Matlab code available in the book Digital Modulations using Matlab

function [dCap]= mpsk_detector(M,r)
%Function to detect MPSK modulated symbols
%[dCap]= mpsk_detector(M,r) detects the received MPSK signal points
%points - 'r'. M is the modulation level of MPSK
   ref_i= 1/sqrt(2)*cos(((1:1:M)-1)/M*2*pi);
   ref_q= 1/sqrt(2)*sin(((1:1:M)-1)/M*2*pi);
   ref = ref_i+1i*ref_q; %reference constellation for MPSK
   [˜,dCap]= iqOptDetector(r,ref); %IQ detection
end

Python code

Full Python code available in the book Digital Modulations using Python

Performance simulation results

The simulation results for error rate performance of M-PSK modulations over AWGN channel and Rayleigh flat-fading channel is given in the following figures.

*Figure 2: Error rate performance of MPSK modulations in AWGN channel*

*Figure 3: Error rate performance of MPSK modulations in Rayleigh flat-fading channel*

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

Reference

[1] John G. Proakis, “Digital Communciations”, McGraw-Hill; 5th edition.↗

Books by the author

Wireless Communication Systems in Matlab Second Edition(PDF) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Python (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Matlab (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart
Hand-picked Best books on Communication Engineering Best books on Signal Processing

Derive BPSK BER – optimum receiver in AWGN channel

Key focus: Derive BPSK BER (bit error rate) for optimum receiver in AWGN channel. Explained intuitively step by step.

BPSK modulation is the simplest of all the M-PSK techniques. An insight into the derivation of error rate performance of an optimum BPSK receiver is essential as it serves as a stepping stone to understand the derivation for other comparatively complex techniques like QPSK,8-PSK etc..

Understanding the concept of Q function and error function is a pre-requisite for this section of article.

The ideal constellation diagram of a BPSK transmission (Figure 1) contains two constellation points located equidistant from the origin. Each constellation point is located at a distance from the origin, where E_s is the BPSK symbol energy. Since the number of bits in a BPSK symbol is always one, the notations – symbol energy (E_s) and bit energy (E_b) can be used interchangeably (E_s=E_b).

Assume that the BPSK symbols are transmitted through an AWGN channel characterized by variance = N₀/2 Watts. When 0 is transmitted, the received symbol is represented by a Gaussian random variable ‘r‘ with mean=S₀ = and variance =N₀/2. When 1 is transmitted, the received symbol is represented by a Gaussian random variable – r with mean=S₁= and variance =N₀/2. Hence the conditional density function of the BPSK symbol (Figure 2) is given by,

*Figure 2: Probability density function (PDF) for BPSK Symbols*

An optimum receiver for BPSK can be implemented using a correlation receiver or a matched filter receiver (Figure 3). Both these forms of implementations contain a decision making block that decides upon the bit/symbol that was transmitted based on the observed bits/symbols at its input.

When the BPSK symbols are transmitted over an AWGN channel, the symbols appears smeared/distorted in the constellation depending on the SNR condition of the channel. A matched filter or that was previously used to construct the BPSK symbols at the transmitter. This process of projection is illustrated in Figure 4. Since the assumed channel is of Gaussian nature, the continuous density function of the projected bits will follow a Gaussian distribution. This is illustrated in Figure 5.

*Figure 4: Role of correlation/Matched Filter*

After the signal points are projected on the basis function axis, a decision maker/comparator acts on those projected bits and decides on the fate of those bits based on the threshold set. For a BPSK receiver, if the a-prior probabilities of transmitted 0’s and 1’s are equal (P=0.5), then the decision boundary or threshold will pass through the origin. If the apriori probabilities are not equal, then the optimum threshold boundary will shift away from the origin.

*Figure 5: Distribution of received symbols*

Considering a binary symmetric channel, where the apriori probabilities of 0’s and 1’s are equal, the decision threshold can be conveniently set to T=0. The comparator, decides whether the projected symbols are falling in region A or region B (see Figure 4). If the symbols fall in region A, then it will decide that 1 was transmitted. It they fall in region B, the decision will be in favor of ‘0’.

For deriving the performance of the receiver, the decision process made by the comparator is applied to the underlying distribution model (Figure 5). The symbols projected on the axis will follow a Gaussian distribution. The threshold for decision is set to T=0. A received bit is in error, if the transmitted bit is ‘0’ & the decision output is ‘1’ and if the transmitted bit is ‘1’ & the decision output is ‘0’.

This is expressed in terms of probability of error as,

Or equivalently,

By applying Bayes Theorem↗, the above equation is expressed in terms of conditional probabilities as given below,

Since a-prior probabilities are equal P(0_T)= P(1_T) =0.5, the equation can be re-written as

Intuitively, the integrals represent the area of shaded curves as shown in Figure 6. From the previous article, we know that the area of the shaded region is given by Q function.

*Figure 6a, 6b: Calculating Error Probability*

Similarly,

From (4), (6), (7) and (8),

For BPSK, since E_s=E_b, the probability of symbol error (P_s) and the probability of bit error (P_b) are same. Therefore, expressing the P_s and P_b in terms of Q function and also in terms of complementary error function :

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

Reference

[1] Nguyen & Shwedyk, “A First course in Digital Communications”, Cambridge University Press, 1st edition.↗

Books by author

Wireless Communication Systems in Matlab Second Edition(PDF) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Python (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Matlab (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart
Hand-picked Best books on Communication Engineering Best books on Signal Processing

Derivation of expression for a Gaussian Filter with 3 dB bandwidth

Note: There is a rating embedded within this post, please visit this post to rate it.
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 f_G(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 probability density function for a Gaussian Distribution with mean=0 and standard deviation=σ is given by

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

Setting f=B,

External Resources:

[1] The care and feeding of digital, pulse-shaping filters – By Ken Gentile
[2] Inter Symbol Interference and Root Raised Cosine Filtering – Complex2real

Recommended Books

Eb/N0 Vs BER for BPSK over Rayleigh Channel and AWGN Channel

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.

This article is part of the following books
● Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885
● Digital Modulations using Python ISBN: 978-1712321638
● Wireless communication systems in Matlab ISBN: 979-8648350779
All books available in ebook (PDF) and Paperback formats

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

$$ \begin{matrix} r=real(\hat{y})=real(x+z) \\ \; \; \hat{d} =1, \; \; if \;r> 0 \\ \; \; \hat{d}=0 , \; \; if \; r< 0 \end{matrix} $$

Theoretical BER:

The theoretical BER for BPSK modulation scheme over Rayleigh fading channel (with AWGN noise) is given by

$$ P_{b} =\frac{1}{2} \left ( 1-\sqrt{\frac{E_{b}/N_{0}}{1+E_{b}/N_{0}}}\right) $$

The theoretical BER for BPSK modulation scheme over an AWGN channel is given here for comparison

$$ P_{b}=\frac{1}{2}erfc(\sqrt{E_{b}/N_{0}}) $$

Simulation Model:

The following model is used for the simulation of BPSK over Rayleigh Fading channel and its comparison with AWGN channel

BPSK Modulation over Rayleigh and AWGN channel

Matlab Code:

Check these books for matlab code

This article is part of the following books
● Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885
● Digital Modulations using Python ISBN: 978-1712321638
● Wireless communication systems in Matlab ISBN: 979-8648350779
All books available in ebook (PDF) and Paperback formats

Simulation Results:

The Simulated and theoretical performance curves (Eb/N0 Vs BER) for BPSK modulation over Rayleigh Fading channel and the AWGN is given below.

Eb/N0 Vs BER for BPSK over Rayleigh and AWGN Channel

Books by the author

Wireless Communication Systems in Matlab Second Edition(PDF) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Python (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Matlab (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart
Hand-picked Best books on Communication Engineering Best books on Signal Processing

External Resources

[1]Theoretical expressions for BER under various conditions
[2]Capacity of MRC on correlated Rician Fading Channels

QPSK – Quadrature Phase Shift Keying

Quadrature Phase Shift Keying (QPSK) is a form of phase modulation technique, in which two information bits (combined as one symbol) are modulated at once, selecting one of the four possible carrier phase shift states.

*Figure 1: Waveform* simulation model for QPSK modulation

The QPSK signal within a symbol duration $T_{sym}$ is defined as

\[s(t) = A \cdot cos \left[2 \pi f_c t + \theta_n \right], \quad \quad 0 \leq t \leq T_{sym},\; n=1,2,3,4 \quad \quad (1) \]

where the signal phase is given by

\[\theta_n = \left(2n – 1 \right) \frac{\pi}{4} \quad \quad (2)\]

Therefore, the four possible initial signal phases are $\pi/4, 3 \pi/4, 5 \pi/4$ and $7 \pi/4$ radians. Equation (1) can be re-written as

\[\begin{align} s(t) &= A \cdot cos \theta_n \cdot cos \left( 2 \pi f_c t\right) – A \cdot sin \theta_n \cdot sin \left( 2 \pi f_c t\right) \\ &= s_{ni} \phi_i(t) + s_{nq} \phi_q(t) \quad\quad \quad \quad \quad\quad \quad \quad \quad\quad \quad \quad \quad\quad \quad \quad (3) \end{align} \]

The above expression indicates the use of two orthonormal basis functions: $ \left\langle \phi_i(t),\phi_q(t)\right\rangle$ together with the inphase and quadrature signaling points: $ \left\langle s_{ni}, s_{nq}\right\rangle$. Therefore, on a two dimensional co-ordinate system with the axes set to $ \phi_i(t)$ and $\phi_q(t)$, the QPSK signal is represented by four constellation points dictated by the vectors $\left\langle s_{ni}, s_{nq}\right\rangle$ with $ n=1,2,3,4$.

The transmitter

The QPSK transmitter, shown in Figure 1, is implemented as a matlab function qpsk_mod. In this implementation, a splitter separates the odd and even bits from the generated information bits. Each stream of odd bits (quadrature arm) and even bits (in-phase arm) are converted to NRZ format in a parallel manner.

Refer Digital Modulations using Matlab : Build Simulation Models from Scratch for full Matlab code.
Refer Digital Modulations using Python for full Python code

File 1: qpsk_mod.m: QPSK modulator

function [s,t,I,Q] = qpsk_mod(a,fc,OF)
%Modulate an incoming binary stream using conventional QPSK
%a - input binary data stream (0's and 1's) to modulate
%fc - carrier frequency in Hertz
%OF - oversampling factor (multiples of fc) - at least 4 is better
%s - QPSK modulated signal with carrier
%t - time base for the carrier modulated signal
%I - baseband I channel waveform (no carrier)
%Q - baseband Q channel waveform (no carrier)
L = 2*OF;%samples in each symbol (QPSK has 2 bits in each symbol)
ak = 2*a-1; %NRZ encoding 0-> -1, 1->+1
I = ak(1:2:end);Q = ak(2:2:end);%even and odd bit streams
I=repmat(I,1,L).'; Q=repmat(Q,1,L).';%even/odd streams at 1/2Tb baud
I = I(:).'; Q = Q(:).'; %serialize
fs = OF*fc; %sampling frequency
t=0:1/fs:(length(I)-1)/fs; %time base
iChannel = I.*cos(2*pi*fc*t);qChannel = -Q.*sin(2*pi*fc*t);
s = iChannel + qChannel; %QPSK modulated baseband signal

The timing diagram for BPSK and QPSK modulation is shown in Figure 2. For BPSK modulation the symbol duration for each bit is same as bit duration, but for QPSK the symbol duration is twice the bit duration: $T_{sym}=2T_b$. Therefore, if the QPSK symbols were transmitted at same rate as BPSK, it is clear that QPSK sends twice as much data as BPSK does. After oversampling and pulse shaping, it is intuitively clear that the signal on the I-arm and Q-arm are BPSK signals with symbol duration $2T_b$. The signal on the in-phase arm is then multiplied by $cos (2 \pi f_c t)$ and the signal on the quadrature arm is multiplied by $-sin (2 \pi f_c t)$. QPSK modulated signal is obtained by adding the signal from both in-phase and quadrature arms.

Note: The oversampling rate for the simulation is chosen as $L=2 f_s/f_c$, where $f_c$ is the given carrier frequency and $f_s$ is the sampling frequency satisfying Nyquist sampling theorem with respect to the carrier frequency ($f_s \geq f_c$). This configuration gives integral number of carrier cycles for one symbol duration.

*Figure 2: Timing diagram for BPSK and QPSK modulations*

The receiver

Due to its special relationship with BPSK, the QPSK receiver takes the simplest form as shown in Figure 3. In this implementation, the I-channel and Q-channel signals are individually demodulated in the same way as that of BPSK demodulation. After demodulation, the I-channel bits and Q-channel sequences are combined into a single sequence. The function qpsk_demod implements a QPSK demodulator as per Figure 3.

Read more about QPSK, implementation of their modulator and demodulator, performance simulation in these books:

*Figure 3: Waveform simulation model for QPSK demodulation*

Performance simulation over AWGN

The complete waveform simulation for the aforementioned QPSK modulation and demodulation is given next. The simulation involves, generating random message bits, modulating them using QPSK modulation, addition of AWGN channel noise corresponding to the given signal-to-noise ratio and demodulating the noisy signal using a coherent QPSK receiver. The waveforms at the various stages of the modulator are shown in the Figure 4.

*Figure 4: Simulated QPSK waveforms at the transmitter side*

The performance simulation for the QPSK transmitter-receiver combination was also coded in the code given above and the resulting bit-error rate performance curve will be same as that of conventional BPSK. A QPSK signal essentially combines two orthogonally modulated BPSK signals. Therefore, the resulting performance curves for QPSK – $E_b/N_0$ Vs. bits-in-error – will be same as that of conventional BPSK.

QPSK variants

QPSK modulation has several variants, three such flavors among them are: Offset QPSK, π/4-QPSK and π/4-DQPSK.

Offset-QPSK

Offset-QPSK is essentially same as QPSK, except that the orthogonal carrier signals on the I-channel and the Q-channel are staggered (one of them is delayed in time). In OQPSK, the orthogonal components cannot change states at the same time, this is because the components change state only at the middle of the symbol periods (due to the half symbol offset in the Q-channel). This eliminates 180° phase shifts all together and the phase changes are limited to 0° or 90° every bit period.

Elimination of 180° phase shifts in OQPSK offers many advantages over QPSK. Unlike QPSK, the spectrum of OQPSK remains unchanged when band-limited [1]. Additionally, OQPSK performs better than QPSK when subjected to phase jitters [2]. Further improvements to OQPSK can be obtained if the phase transitions are avoided altogether – as evident from continuous modulation schemes like Minimum Shift Keying (MSK) technique.

π/4-QPSK and π/4-DQPSK

In π/4-QPSK, the signaling points of the modulated signals are chosen from two QPSK constellations that are just shifted π/4 radians (45°) with respect to each other. Switching between the two constellations every successive bit ensures that the phase changes are confined to odd multiples of 45°. Therefore, phase transitions of 90° and 180° are eliminated.

π/4-QPSK preserves the constant envelope property better than QPSK and OQPSK. Unlike QPSK and OQPSK schemes, π/4-QPSK can be differentially encoded, therefore enabling the use of both coherent and non-coherent demodulation techniques. Choice of non-coherent demodulation results in simpler receiver design. Differentially encoded π/4-QPSK is referred as π/4-DQPSK.

Read more about QPSK and its variants, implementation of their modulator and demodulator, performance simulation in these books:

Constellation diagram

The phase transition properties of the different variants of QPSK schemes, are easily investigated using constellation diagram. Refer this article that discusses the method to plot signal space constellations, for the various modulations used in the transmitter.

*Figure 5: Constellations plots for: (a) a = 0.3 RC-filtered QPSK, (b) α = 0.3 RC-filtered O-QPSK, (c) α = 0.3 RC-filtered π/4-DQPSK and (d) MSK*

Refer Digital Modulations using Matlab : Build Simulation Models from Scratch for full Matlab code.
Refer Digital Modulations using Python for full Python code

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

References

[1] S. A. Rhodes, “Effects of hardlimiting on bandlimited transmissions with conventional and offset QPSK modulation”, in Proc. Nat. TeIecommun. Conf., Houston, TX, 1972, PP. 20F/1-20F/7
[2] S. A. Rhodes, “Effect of noisy phase reference on coherent detection of offset QPSK signals”, IEEE Trans. Commun., vol. COM-22, PP. 1046-1055, Aug. 1974.↗

Books by the author

Wireless Communication Systems in Matlab Second Edition(PDF) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Python (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Matlab (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart
Hand-picked Best books on Communication Engineering Best books on Signal Processing

Topics in this chapter

Digital Modulators and Demodulators - Passband Simulation Models
● Introduction
● Binary Phase Shift Keying (BPSK)
□ BPSK transmitter
□ BPSK receiver
□ End-to-end simulation
● Coherent detection of Differentially Encoded BPSK (DEBPSK)
● Differential BPSK (D-BPSK)
□ Sub-optimum receiver for DBPSK
□ Optimum noncoherent receiver for DBPSK
● Quadrature Phase Shift Keying (QPSK)
□ QPSK transmitter
□ QPSK receiver
□ Performance simulation over AWGN
● Offset QPSK (O-QPSK)
● π/p=4-DQPSK
● Continuous Phase Modulation (CPM)
□ Motivation behind CPM
□ Continuous Phase Frequency Shift Keying (CPFSK) modulation
□ Minimum Shift Keying (MSK)
● Investigating phase transition properties
● Power Spectral Density (PSD) plots
● Gaussian Minimum Shift Keying (GMSK)
□ Pre-modulation Gaussian Low Pass Filter
□ Quadrature implementation of GMSK modulator
□ GMSK spectra
□ GMSK demodulator
□ Performance
● Frequency Shift Keying (FSK)
□ Binary-FSK (BFSK)
□ Orthogonality condition for non-coherent BFSK detection
□ Orthogonality condition for coherent BFSK
□ Modulator
□ Coherent Demodulator
□ Non-coherent Demodulator
□ Performance simulation
□ Power spectral density

Performance comparison of Digital Modulation techniques

Key focus: Compare Performance and spectral efficiency of bandwidth-efficient digital modulation techniques (BPSK,QPSK and QAM) on their theoretical BER over AWGN.

More detailed analysis of Shannon’s theorem and Channel capacity is available in the following book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

Simulation of various digital modulation techniques are available in these books
● Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885
● Digital Modulations using Python ISBN: 978-1712321638

Let’s take up some bandwidth-efficient linear digital modulation techniques (BPSK,QPSK and QAM) and compare its performance based on their theoretical BER over AWGN. (Readers are encouraged to read previous article on Shannon’s theorem and channel capacity).

Table 1 summarizes the theoretical BER (given SNR per bit ration – E_b/N₀) for various linear modulations. Note that the E_b/N₀ values used in that table are in linear scale [to convert E_b/N₀ in dB to linear scale – use E_b/N₀(linear) = 10^(E_b/N₀(dB)/10) ]. A small script written in Matlab (given below) gives the following output.

*Figure 1: Eb/N0 Vs. BER for various digital modulations over AWGN channel*

*Table 1: Theoretical BER over AWGN for various linear digital modulation techniques*

The following table is obtained by extracting the values of Eb/N0 to achieve BER=10^-6 from Figure-1. (Table data sorted with increasing values of E_b/N₀).

*Table 2: Capacity of various modulations their efficiency and channel bandwidth*

where,

is the bandwidth efficiency for linear modulation with M point constellation, meaning that η_Bbits can be stuffed in one symbol with R_b bits/sec data rate for a given minimum bandwidth.

is the minimum bandwidth needed for information rate of R_b bits/second. If a pulse shaping technique like raised cosine pulse [with roll off factor (a)] is used then B_min becomes

Next the data in table 2 is plotted with E_b/N₀ on the x-axis and η on the y-axis (see figure 2) along with the well known Shannon’s Capacity equation over AWGN given by,

which can be represented as (refer [1])

*Figure 2: Spectral efficiency vs Eb/N0 for various modulations at P_b=10^-6*

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

Matlab Code

EbN0dB=-4:1:24;
EbN0lin=10.^(EbN0dB/10);
colors={'b-*','g-o','r-h','c-s','m-d','y-*','k-p','b--&gt;','g:&lt;','r-.d'};
index=1;

%BPSK
BPSK = 0.5*erfc(sqrt(EbN0lin));
plotHandle=plot(EbN0dB,log10(BPSK),char(colors(index)));
set(plotHandle,'LineWidth',1.5);
hold on;

index=index+1;

%M-PSK
m=2:1:5;
M=2.^m;
for i=M,
    k=log2(i);
    berErr = 1/k*erfc(sqrt(EbN0lin*k)*sin(pi/i));
    plotHandle=plot(EbN0dB,log10(berErr),char(colors(index)));
    set(plotHandle,'LineWidth',1.5);
    index=index+1;
end

%Binary DPSK
Pb = 0.5*exp(-EbN0lin);
plotHandle = plot(EbN0dB,log10(Pb),char(colors(index)));
set(plotHandle,'LineWidth',1.5);
index=index+1;

%Differential QPSK
a=sqrt(2*EbN0lin*(1-sqrt(1/2)));
b=sqrt(2*EbN0lin*(1+sqrt(1/2)));
Pb = marcumq(a,b,1)-1/2.*besseli(0,a.*b).*exp(-1/2*(a.^2+b.^2));
plotHandle = plot(EbN0dB,log10(Pb),char(colors(index)));
set(plotHandle,'LineWidth',1.5);
index=index+1;

%M-QAM
m=2:2:6;
M=2.^m;

for i=M,
    k=log2(i);
    berErr = 2/k*(1-1/sqrt(i))*erfc(sqrt(3*EbN0lin*k/(2*(i-1))));
    plotHandle=plot(EbN0dB,log10(berErr),char(colors(index)));
    set(plotHandle,'LineWidth',1.5);
    index=index+1;
end

legend('BPSK','QPSK','8-PSK','16-PSK','32-PSK','D-BPSK','D-QPSK','4-QAM','16-QAM','64-QAM');
axis([-4 24 -8 0]);
set(gca,'XTick',-4:2:24); %re-name axis accordingly
ylabel('Probability of BER Error - log10(Pb)');
xlabel('Eb/N0 (dB)');
title('Probability of BER Error log10(Pb) Vs Eb/N0');
grid on;

Reference

[1] “Digital Communications” by John G.Proakis ,Chapter 7: Channel Capacity and Coding.↗

Books by the author

Wireless Communication Systems in Matlab Second Edition(PDF) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Python (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Matlab (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart
Hand-picked Best books on Communication Engineering Best books on Signal Processing

BPSK – Binary Phase Shift Keying

Key focus: BPSK, Binary Phase Shift Keying, bpsk modulation, bpsk demodulation, BPSK matlab, BPSK python implementation, BPSK constellation

BPSK – introduction

BPSK stands for Binary Phase Shift Keying. It is a type of modulation used in digital communication systems to transmit binary data over a communication channel.

In BPSK, the carrier signal is modulated by changing its phase by 180 degrees for each binary symbol. Specifically, a binary 0 is represented by a phase shift of 180 degrees, while a binary 1 is represented by no phase shift.

BPSK is a straightforward and effective modulation method and is frequently utilized in applications where the communication channel is susceptible to noise and interference. It is also utilized in different wireless communication systems like Wi-Fi, Bluetooth, and satellite communication.

Implementation details

Binary Phase Shift Keying (BPSK) is a two phase modulation scheme, where the 0’s and 1’s in a binary message are represented by two different phase states in the carrier signal: $\theta=0^{\circ}$ for binary 1 and $\theta=180^{\circ}$ for binary 0.

In digital modulation techniques, a set of basis functions are chosen for a particular modulation scheme. Generally, the basis functions are orthogonal to each other. Basis functions can be derived using Gram Schmidt orthogonalization procedure [1]. Once the basis functions are chosen, any vector in the signal space can be represented as a linear combination of them. In BPSK, only one sinusoid is taken as the basis function. Modulation is achieved by varying the phase of the sinusoid depending on the message bits. Therefore, within a bit duration $T_b$, the two different phase states of the carrier signal are represented as,

\begin{align*} s_1(t) &= A_c\; cos\left(2 \pi f_c t \right), & 0 \leq t \leq T_b \quad \text{for binary 1}\\ s_0(t) &= A_c\; cos\left(2 \pi f_c t + \pi \right), & 0 \leq t \leq T_b \quad \text{for binary 0} \end{align*}

where, $A_c$ is the amplitude of the sinusoidal signal, $f_c$ is the carrier frequency $Hz$, $t$ being the instantaneous time in seconds, $T_b$ is the bit period in seconds. The signal $s_0(t)$ stands for the carrier signal when information bit $a_k=0$ was transmitted and the signal $s_1(t)$ denotes the carrier signal when information bit $a_k=1$ was transmitted.

The constellation diagram for BPSK (Figure 3 below) will show two constellation points, lying entirely on the x axis (inphase). It has no projection on the y axis (quadrature). This means that the BPSK modulated signal will have an in-phase component but no quadrature component. This is because it has only one basis function. It can be noted that the carrier phases are $180^{\circ}$ apart and it has constant envelope. The carrier’s phase contains all the information that is being transmitted.

BPSK transmitter

A BPSK transmitter, shown in Figure 1, is implemented by coding the message bits using NRZ coding ($1$ represented by positive voltage and $0$ represented by negative voltage) and multiplying the output by a reference oscillator running at carrier frequency $f_c$.

The following function (bpsk_mod) implements a baseband BPSK transmitter according to Figure 1. The output of the function is in baseband and it can optionally be multiplied with the carrier frequency outside the function. In order to get nice continuous curves, the oversampling factor ($L$) in the simulation should be appropriately chosen. If a carrier signal is used, it is convenient to choose the oversampling factor as the ratio of sampling frequency ($f_s$) and the carrier frequency ($f_c$). The chosen sampling frequency must satisfy the Nyquist sampling theorem with respect to carrier frequency. For baseband waveform simulation, the oversampling factor can simply be chosen as the ratio of bit period ($T_b$) to the chosen sampling period ($T_s$), where the sampling period is sufficiently smaller than the bit period.

Refer Digital Modulations using Matlab : Build Simulation Models from Scratch for full Matlab code.
Refer Digital Modulations using Python for full Python code

File 1: bpsk_mod.m: Baseband BPSK modulator

function [s_bb,t] = bpsk_mod(ak,L)
%Function to modulate an incoming binary stream using BPSK(baseband)
%ak - input binary data stream (0's and 1's) to modulate
%L - oversampling factor (Tb/Ts)
%s_bb - BPSK modulated signal(baseband)
%t - generated time base for the modulated signal
N = length(ak); %number of symbols
a = 2*ak-1; %BPSK modulation
ai=repmat(a,1,L).'; %bit stream at Tb baud with rect pulse shape
ai = ai(:).';%serialize
t=0:N*L-1; %time base
s_bb = ai;%BPSK modulated baseband signal

BPSK receiver

A correlation type coherent detector, shown in Figure 2, is used for receiver implementation. In coherent detection technique, the knowledge of the carrier frequency and phase must be known to the receiver. This can be achieved by using a Costas loop or a Phase Lock Loop (PLL) at the receiver. For simulation purposes, we simply assume that the carrier phase recovery was done and therefore we directly use the generated reference frequency at the receiver – $cos( 2 \pi f_c t)$.

*Figure 2: Coherent detection of BPSK (correlation type)*

In the coherent receiver, the received signal is multiplied by a reference frequency signal from the carrier recovery blocks like PLL or Costas loop. Here, it is assumed that the PLL/Costas loop is present and the output is completely synchronized. The multiplied output is integrated over one bit period using an integrator. A threshold detector makes a decision on each integrated bit based on a threshold. Since, NRZ signaling format was used in the transmitter, the threshold for the detector would be set to $0$. The function bpsk_demod, implements a baseband BPSK receiver according to Figure 2. To use this function in waveform simulation, first, the received waveform has to be downconverted to baseband, and then the function may be called.

Refer Digital Modulations using Matlab : Build Simulation Models from Scratch for full Matlab code.
Refer Digital Modulations using Python for full Python code

File 2: bpsk_demod.m: Baseband BPSK detection (correlation receiver)

function [ak_cap] = bpsk_demod(r_bb,L)
%Function to demodulate an BPSK(baseband) signal
%r_bb - received signal at the receiver front end (baseband)
%N - number of symbols transmitted
%L - oversampling factor (Tsym/Ts)
%ak_cap - detected binary stream
x=real(r_bb); %I arm
x = conv(x,ones(1,L));%integrate for L (Tb) duration
x = x(L:L:end);%I arm - sample at every L
ak_cap = (x > 0).'; %threshold detector

End-to-end simulation

The complete waveform simulation for the end-to-end transmission of information using BPSK modulation is given next. The simulation involves: generating random message bits, modulating them using BPSK modulation, addition of AWGN noise according to the chosen signal-to-noise ratio and demodulating the noisy signal using a coherent receiver. The topic of adding AWGN noise according to the chosen signal-to-noise ratio is discussed in section 4.1 in chapter 4. The resulting waveform plots are shown in the Figure 2.3. The performance simulation for the BPSK transmitter/receiver combination is also coded in the program shown next (see chapter 4 for more details on theoretical error rates).

The resulting performance curves will be same as the ones obtained using the complex baseband equivalent simulation technique in Figure 4.4 of chapter 4.

Refer Digital Modulations using Matlab : Build Simulation Models from Scratch for full Matlab code.
Refer Digital Modulations using Python for full Python code

File 3: bpsk_wfm_sim.m: Waveform simulation for BPSK modulation and demodulation

*Figure 3: (a) Baseband BPSK signal,(b) transmitted BPSK signal – with carrier, (c) constellation at transmitter and (d) received signal with AWGN noise*

References:

[1] Lloyd N. Trefethen, David Bau III , Numerical linear algebra, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-361-9, pp.56

Books by the author

Wireless Communication Systems in Matlab Second Edition(PDF) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Python (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Matlab (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart
Hand-picked Best books on Communication Engineering Best books on Signal Processing

Cookie	Duration	Description
cookielawinfo-checbox-analytics	11 months	This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Analytics".
cookielawinfo-checbox-analytics	11 months	This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Analytics".
cookielawinfo-checbox-functional	11 months	The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional".
cookielawinfo-checbox-functional	11 months	The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional".
cookielawinfo-checbox-others	11 months	This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Other.
cookielawinfo-checbox-others	11 months	This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Other.
cookielawinfo-checkbox-necessary	11 months	This cookie is set by GDPR Cookie Consent plugin. The cookies is used to store the user consent for the cookies in the category "Necessary".
cookielawinfo-checkbox-performance	11 months	This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Performance".
viewed_cookie_policy	11 months	The cookie is set by the GDPR Cookie Consent plugin and is used to store whether or not user has consented to the use of cookies. It does not store any personal data.

Rectangular QAM from PAM constellation

Matlab code

Python code

M-QAM demodulation (coherent detection)

Matlab code

Python code

Performance simulation results

Reference

Related Topics

Books by the author

M-ary phase shift keying (M-PSK) modulation

Matlab code

Python code

M-PSK demodulation (coherent detection)

Matlab code

Python code

Performance simulation results

Reference

Related Topics

Books by the author

Reference

Books by author

See also :

External Resources:

Recommended Books

Non-Coherent Detection:

Coherent Detection:

Theoretical BER:

Simulation Model:

Matlab Code:

Simulation Results:

See also

Books by the author

External Resources

The transmitter

The receiver

Performance simulation over AWGN

QPSK variants

Offset-QPSK

π/4-QPSK and π/4-DQPSK

Constellation diagram

References

Books by the author

Topics in this chapter

Matlab Code

Reference

Related topics

Books by the author

BPSK – introduction

Implementation details

BPSK transmitter

BPSK receiver

End-to-end simulation

References:

Books by the author

Topics in this chapter