Draw the autocorrelation function for the following codes: 1- 0010111 2- 0000000 Find which code has a better correlation function and why.
The correct answer and explanation is:
To find the autocorrelation function for the two binary codes and determine which has better autocorrelation properties, we follow these steps:
Step 1: Convert binary to bipolar format
We replace:
0→-11→+1
Code 1: 0010111 → Bipolar: -1 -1 +1 -1 +1 +1 +1
Code 2: 0000000 → Bipolar: -1 -1 -1 -1 -1 -1 -1
Step 2: Compute autocorrelation function (ACF)
The autocorrelation function R(k)R(k) of a sequence x(n)x(n) is calculated as: R(k)=∑n=0N−k−1x(n)⋅x(n+k)R(k) = \sum_{n=0}^{N-k-1} x(n) \cdot x(n+k)
Where kk is the lag (0 to N-1).
Let’s compute the autocorrelation functions manually for both codes.
Code 1: ACF of 0010111 (bipolar)
Let x=[−1,−1,+1,−1,+1,+1,+1]x = [-1, -1, +1, -1, +1, +1, +1]
- R(0)=(−1)2+(−1)2+12+(−1)2+12+12+12=7R(0) = (-1)^2 + (-1)^2 + 1^2 + (-1)^2 + 1^2 + 1^2 + 1^2 = 7
- R(1)=(−1)(−1)+(−1)(1)+(1)(−1)+(−1)(1)+(1)(1)+(1)(1)=1R(1) = (-1)(-1) + (-1)(1) + (1)(-1) + (-1)(1) + (1)(1) + (1)(1) = 1
- R(2)=(−1)(1)+(−1)(−1)+(1)(1)+(−1)(1)+(1)(1)=1R(2) = (-1)(1) + (-1)(-1) + (1)(1) + (-1)(1) + (1)(1) = 1
- R(3)=(−1)(−1)+(−1)(1)+(1)(1)+(−1)(1)=2R(3) = (-1)(-1) + (-1)(1) + (1)(1) + (-1)(1) = 2
- R(4)=(−1)(1)+(−1)(1)+(1)(1)=−1R(4) = (-1)(1) + (-1)(1) + (1)(1) = -1
- R(5)=(−1)(1)+(−1)(1)=−2R(5) = (-1)(1) + (-1)(1) = -2
- R(6)=(−1)(1)=−1R(6) = (-1)(1) = -1
So, ACF = [7, 1, 1, 2, -1, -2, -1]
Code 2: ACF of 0000000 (bipolar)
Let x=[−1,−1,−1,−1,−1,−1,−1]x = [-1, -1, -1, -1, -1, -1, -1]
- R(0)=7R(0) = 7
- R(1)=(−1)2+(−1)2+…=6R(1) = (-1)^2 + (-1)^2 + … = 6
- R(2)=5R(2) = 5
- …
- R(6)=1R(6) = 1
So, ACF = [7, 6, 5, 4, 3, 2, 1]
Analysis & Conclusion (300 Words)
Autocorrelation is a measure of how well a signal matches a shifted version of itself. A good autocorrelation function for communication systems has a sharp peak at lag 0 and low values (ideally zero) elsewhere. This property helps in distinguishing the signal from noise and in detecting signal timing accurately.
From the results above:
- Code 1 has an ACF of: [7, 1, 1, 2, -1, -2, -1]
- Code 2 has an ACF of: [7, 6, 5, 4, 3, 2, 1]
While both codes have the same peak value at lag 0 (which is expected, as this represents total signal energy), Code 1 shows much lower side-lobes, and some values are even negative or close to zero. These lower values away from the peak (lag 0) mean that Code 1 exhibits better autocorrelation properties.
In contrast, Code 2 is a constant sequence (all zeros), leading to high correlation at every lag. This uniformity makes it poor for signal detection because it lacks uniqueness — the receiver cannot easily distinguish the start of the code, increasing the risk of synchronization errors.
Thus, Code 1 is better, as it has a distinct, narrow peak and relatively low off-peak values — ideal for minimizing cross-talk and ensuring accurate detection in digital communication systems like CDMA and spread-spectrum.
✅ Correct Answer: Code 1 (0010111) has a better correlation function because it has lower off-peak autocorrelation values, which improves signal detection and synchronization.