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**Digital Communications I: Modulation and Coding Course**

Spring Jeffrey N. Denenberg Lecture 4b: Detection of M-ary Bandpass Signals

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**Last time we talked about:**

Some bandpass modulation schemes M-PAM, M-PSK, M-FSK, M-QAM How to perform coherent and non- coherent detection Lecture 8

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**Example of two dim. modulation**

“110” “000” “001” “011” “010” “101” “111” “100” 16QAM “0000” “0001” “0011” “0010” 1 3 -1 -3 “1000” “1001” “1011” “1010” “1100” “1101” “1111” “1110” “0100” “0101” “0111” “0110” 8PSK “00” “11” “10” “01” QPSK Lecture 8

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**Today, we are going to talk about:**

How to calculate the average probability of symbol error for different modulation schemes that we studied? How to compare different modulation schemes based on their error performances? Lecture 8

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**Error probability of bandpass modulation**

Before evaluating the error probability, it is important to remember that: The type of modulation and detection ( coherent or non- coherent) determines the structure of the decision circuits and hence the decision variable, denoted by z. The decision variable, z, is compared with M-1 thresholds, corresponding to M decision regions for detection purposes. Decision Circuits Compare z with threshold. Lecture 8

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Error probability … The matched filters output (observation vector= ) is the detector input and the decision variable is a function of , i.e. For MPAM, MQAM and MFSK with coherent detection For MPSK with coherent detection For non-coherent detection (M-FSK and DPSK), We know that for calculating the average probability of symbol error, we need to determine Hence, we need to know the statistics of z, which depends on the modulation scheme and the detection type. Lecture 8

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**Error probability … AWGN channel model:**

The signal vector is deterministic. The elements of the noise vector are i.i.d Gaussian random variables with zero-mean and variance The noise vector's pdf is The elements of the observed vector are independent Gaussian random variables. Its pdf is Lecture 8

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**Error probability … BPSK and BFSK with coherent detection: Lecture 8**

“0” “1” “0” “1” BPSK BFSK Lecture 8

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**Error probability … Non-coherent detection of BFSK Lecture 8**

Decision variable: Difference of envelopes + - Decision rule: Lecture 8

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**Error probability – cont’d**

Non-coherent detection of BFSK … Similarly, non-coherent detection of DBPSK Rician pdf Rayleigh pdf Lecture 8

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**(Compare with M-1 thresholds)**

Error probability …. Coherent detection of M-PAM Decision variable: “00” “01” “11” “10” 4-PAM ML detector (Compare with M-1 thresholds) Lecture 8

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**Error probability …. Coherent detection of M-PAM …. Lecture 8**

Error happens if the noise, , exceeds in amplitude one-half of the distance between adjacent symbols. For symbols on the border, error can happen only in one direction. Hence: Gaussian pdf with zero mean and variance Lecture 8

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**Error probability … Coherent detection of M-QAM Lecture 8 ML detector**

“0000” “0001” “0011” “0010” “1000” “1001” “1011” “1010” “1100” “1101” “1111” “1110” “0100” “0101” “0111” “0110” 16-QAM ML detector Parallel-to-serial converter Lecture 8

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**Average probability of**

Error probability … Coherent detection of M-QAM … M-QAM can be viewed as the combination of two modulations on I and Q branches, respectively. No error occurs if no error is detected on either the I or the Q branch. Considering the symmetry of the signal space and the orthogonality of the I and Q branches: Average probability of symbol error for Lecture 8

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**Error probability … Coherent detection of MPSK Lecture 8 Compute**

“110” “000” “001” “011” “010” “101” “111” “100” 8-PSK Compute Choose smallest Decision variable Lecture 8

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**Error probability … Coherent detection of MPSK … Lecture 8**

The detector compares the phase of observation vector to M-1 thresholds. Due to the circular symmetry of the signal space, we have: where It can be shown that or Lecture 8

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**Error probability … Coherent detection of M-FSK Lecture 8 ML detector:**

Choose the largest element in the observed vector Lecture 8

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**Error probability … Coherent detection of M-FSK …**

The dimension of the signal space is M. An upper bound for the average symbol error probability can be obtained by using the union bound. Hence: or, equivalently Lecture 8

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**Bit error probability versus symbol error probability**

Number of bits per symbol For orthogonal M-ary signaling (M-FSK) For M-PSK, M-PAM and M-QAM Lecture 8

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**Probability of symbol error for binary modulation**

Note! “The same average symbol energy for different sizes of signal space” Lecture 8

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**Probability of symbol error for M-PSK**

Note! “The same average symbol energy for different sizes of signal space” Lecture 8

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**Probability of symbol error for M-FSK**

Note! “The same average symbol energy for different sizes of signal space” Lecture 8

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**Probability of symbol error for M-PAM**

Note! “The same average symbol energy for different sizes of signal space” Lecture 8

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**Probability of symbol error for M-QAM**

Note! “The same average symbol energy for different sizes of signal space” Lecture 8

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**Example of samples of matched filter output for some bandpass modulation schemes**

Lecture 8

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