Krishna Sankar, Author at DSP LOG

Peak to Average Power Ratio for OFDM

Let us try to understand peak to average power ratio (PAPR) and its typical value in an OFDM system specified per IEEE 802.11a specifications.

What is PAPR?

The peak to average power ratio for a signal $x(t)$ is defined as
$papr = \frac{\max\left[x(t)x^*(t)\right]}{E\left[x(t)x^*(t)\right]}$ , where
$()^*$ corresponds to the conjugate operator.

Expressing in deciBels,
$papr_{dB} = 10\log_{10}(papr)$ .

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Cylcic prefix in Orthogonal Frequency Division Multiplexing

In a previous post (here), we discussed in brief, Orthogonal Frequency Division Multiplexing (OFDM) transmission. Let us know probe bit more into the motivation of cyclic prefix (aka guard interval) associated with each OFDM symbol.

What is cyclic prefix?
Let us consider one subcarrier (subcarrier +1 specified in IEEE 802.11a specification) alone. In the figure shown below, the blue line corresponds to the original sinusoidal where one cycle of the sinusoidal is of duration 64 samples ( $3.2\mu s$ with 20MHz sampling), corresponding to subcarrier of frequency 312.5kHz.

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Understanding an OFDM transmission

Let us try to understand simulation of a typical Orthogonal Frequency Division Multiplexing (OFDM) transmission defined per IEEE 802.11a specification.

Orthogonal pulses
In a previous post (here ), we have understood that the minimum frequency separation for two sinusoidals with arbitrary phases to be orthogonal is $\frac{1}{T}$ , where $T$ is the symbol period.

In Orthogonal Frequency Division Multiplexing, multiple sinusoidals with frequency separation $\frac{1}{T}$ is used. The sinusoidals used in OFDM can be defined as (Refer Sec6.4.2 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]:

$g_k(t) = \frac{1}{\sqrt{T}}e^{\frac{j2\pi kt}{T}}w(t)$ , where

$k=0,1,\ldots,K-1$ correspond to the frequency of the sinusoidal and

$w(t)=u(t)-u(t-T)$ is a rectangular window over $[0\ T)$ .

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Simulating Minimum Shift Keying Transmitter

Minimum shift keying (MSK) is an important concept to learn in digital communications. It is a form of continuous phase frequency shift keying . In minimum phase shift keying, two key concepts are used.

(a) The frequency separation of the sinusoidals used for representing bits 1’s and 0’s are $\frac{1}{2T}$ , where $T$ is the symbol period.

(b) It is ensured that the resulting waveform is phase continuous.

Motivation of continuous phase

In a previous post (here), we have understood that the minimum frequency separation for two sinusoidals having zero phase difference to be orthogonal is $\frac{1}{2T}$ , where $T$ is the symbol period. However, it can be observed that at each symbol boundary, there is a phase discontinuity. The presence of phase discontinuities can result in large spectral side lobes outside the desired bandwidth. Hence the need for having a frequency modulated signal which is phase continuous.
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Minimum frequency spacing for having orthogonal sinusoidals

In this post, the objective is to figure out the minimum separation between two sinusoidals having frequencies $f_1$ , $f_2$ of duration $T$ each to be orthogonal. Let the phase difference between the sinusoidals is $\phi$ where $\phi$ can take any value from $0$ to $2\pi$ (Refer Example 4.3 [DIG-COMM-SKLAR]).

For the two sinuosidals to be orthogonal,

$\int_0^Tcos(2\pi f_1 t+\phi)cos(2 \pi f_2t)dt = 0$

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Using CORDIC for phase and magnitude computation

In a previous post (here), we looked at using CORDIC (Co-ordinate Rotation by DIgital Computer) for understanding how a complex number $\mathbf{X}$ can be rotated by an angle $\theta$ without using actual multipliers. Let us know try to understand how we can use CORDIC for finding the phase and magnitude of a complex number.

Basics

The CORDIC algorithm is built on successively multiplying the complex number $\mathbf{X}$ , by $\mathbf{Y} =1 + j2^{-k}$ . As can be noticed, as the elements of $\mathbf{Y}$ can be represented in powers of 2, the multiplication can be achieved by using the appropriate ‘bit shift’. For further details, please refer to the previous post (CORDIC for phase rotation).

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Symbol Error Rate (SER) for 16-QAM

Given that we have went over the symbol error probability for 4-PAM and symbol error rate for 4-QAM , let us extend the understanding to find the symbol error probability for 16-QAM (16 Quadrature Amplitude Modulation). Consider a typical 16-QAM modulation scheme where the alphabets (Refer example 5-37 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]).

$\alpha_{16QAM}=\left{\pm 1+\pm 1j,\ \pm 1+\pm 3j,\\\pm 3 + \pm 3j,\ \pm 3+\pm 1j \right}$ are used.

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Digital implementation of RC low pass filter

Thanks to the nice article from Xilinx TechXclusives [XLNX-TECH], let us try to understand the probable digital implementation of resistor-capacitor based low pass filter. Consider a simple RC filter shown in the figure below. Assuming that there is no load across the capacitor, the capacitor charges and discharges through the resistor path.

Figure: RC low pass filter

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Symbol Error Rate (SER) for QPSK (4-QAM) modulation

Given that we have discussed symbol error rate probability for a 4-PAM modulation, let us know focus on finding the symbol error probability for a QPSK (4-QAM) modulation scheme.

Background

Consider that the alphabets used for a QPSK (4-QAM) is $\alpha_{QPSK}=\left{\pm 1\ \pm 1j\right}$ (Refer example 5-35 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]).

Download free e-Book discussing theoretical and simulated error rates for the digital modulation schemes like BPSK, QPSK, 4–PAM, 16PSK and 16QAM. Further, Bit Error Rate with Gray coded mapping, bit error rate for BPSK over OFDM are also discussed.
Interested in MIMO (Multiple Input Multiple Output) communications? Click here to see the post describing six equalizers with 2×2 V-BLAST.
Read about using multiple antennas at the transmitter and receiver to improve the diversity of a communication link. Articles include Selection diversity, Equal Gain Combining, Maximal Ratio Combining, Alamouti STBC, Transmit Beaforming.

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Symbol Error Rate (SER) for 4-PAM

Following discussion of bit error rate (BER) for BPSK and bit error rate for FSK, it may be interesting to move on to discuss a higher order constellation such as Pulse Amplitude Modulation (PAM).

Consider that the alphabets used for a 4-PAM is $\alpha_{4PAM}=\left{\pm 1,\ \pm 3\right}$ (Refer example 5-34 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]).

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Coherent demodulation of DBPSK

In a previous post, we discussed about a probable first order digital PLL for tracking constant phase offset. The assumption was that as the phase offset is small and the bits gets decoded correctly, the phase difference between the ideal and actual constellation gives the initial value of phase. However, in typical scenarios it may be possible that the above assumption may not be valid, resulting in phase ambiguity.

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Scaling factor in QAM

When QAM (Quadrature Amplitude Modulation) is used, typically one may find a scaling factor associated with the constellation mapping operation. It may be reasonably obvious that this scaling factor is for normalizing the average energy to one.

This post attempts to compute the average energy of the 16-QAM, 64-QAM and M-QAM constellation (where $\sqrt{M}$ is a power of 2), thanks to the nice example 5.16 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT].

Consider a typical 16-QAM modulation scheme where the alphabets

$\alpha_{16QAM}=\left{\pm 1+\pm 1j,\ \pm 1+\pm 3j,\\\pm 3 + \pm 3j,\ \pm 3+\pm 1j \right}$ are used.

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Bit Error Rate (BER) for frequency shift keying with coherent demodulation

Following the request by Siti Naimah, this post discuss the bit error probability for coherent demodulation of binary Frequency Shift Keying (BFSK) along with a small Matlab code snippet.

Using the definition provided in Sec 4.4.4 of [DIG-COMM-SKLAR]), in binary Frequency shift keying (BFSK), the bits 0’s and 1’s are represented by signals $s_1$ and $s_2$ having frequencies $f_1$ and $f_2$ respectively, i.e.