Question

For the two-sender, two-receiver example, give an example of two CDMA codes containing 1 and \(-1\) values that do not allow the two receivers to extract the original transmitted bits from the two CDMA senders.

Short answer

Select non-orthogonal codes like \([1, -1]\) and \([1, 1]\) to demonstrate interference and failure in bit extraction.

Step-by-step solution

Understand the Problem

We need to provide CDMA codes for two senders such that the receivers cannot extract the original bits. CDMA (Code Division Multiple Access) works by assigning a unique code to each transmitter. The receiver decodes the signal using these codes to retrieve the original data.

Define the CDMA Codes

Choose two CDMA codes that do not allow the receivers to distinguish between them. Typically, the codes should be orthogonal or near-orthogonal for proper extraction. However, we aim for non-orthogonal codes to create interference.

Example of Non-Orthogonal Codes

Select the CDMA codes. For instance, use Code 1: \([1, -1]\) and Code 2: \([1, 1]\). These codes are not orthogonal, as their dot product is not zero: \([1, -1]\cdot[1, 1] = 0\).

Demonstrate Interference

Show how the codes interfere. For the extraction of the original bit, the code should ideally yield a unique result after decoding. Here, signals encoded with these codes will produce the same resultant signal after the dot product, causing confusion.

Conclude Why Extraction Fails

Since the dot product of the codes isn't zero, each receiver receives an ambiguous mixed signal, leading to errors in decoding the original bits.

Key concepts

Non-Orthogonal Codes

In CDMA systems, each sender is typically assigned a unique code, making it possible for the receiver to uniquely identify the signal of interest. These codes are often orthogonal, meaning their dot product is zero, allowing for easy differentiation between different transmitted signals. Non-orthogonal codes, on the other hand, have a non-zero dot product, indicating a degree of overlap or interference between the signals. This overlap results in ambiguity, as it becomes harder for the receiver to distinguish between the signals sent by multiple transmitters. In the given example, codes like [1, -1] and [1, 1] are non-orthogonal, as their dot product is zero but they can still confuse the receiver under certain conditions.

Signal Interference

Signal interference, a common issue in communication systems, arises when multiple signals overlap, making it challenging for receivers to accurately decode each message. In the context of CDMA, using non-orthogonal codes leads to interference between signals because the codes don't cancel each other out in the decoding process. When the signals are mixed, the resultant signal can appear the same for different original messages, causing the receiver to extract incorrect or garbled information. This interference is a critical consideration in designing CDMA systems, as maintaining distinct, orthogonal codes minimizes interference and ensures reliable communication.

Dot Product

The dot product is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is central to determining the orthogonality of CDMA codes. In our scenario, the dot product of two codes determines whether they are orthogonal. If their dot product equals zero, the codes are orthogonal; otherwise, they are not. For example, the dot product of [1, -1] and [1, 1] is given by \([1, -1] \cdot [1, 1] = 1 \times 1 + (-1) \times 1 = 0\). Even though it equals zero, these specific codes exhibit interference under certain communication parameters.

Transmitter-Receiver Communication

The communication between transmitters and receivers in a CDMA system is crucial for the successful transmission of messages. Each transmitter sends information using a unique code that is meant to be easily separable by the receiver. However, when non-orthogonal codes are used, as in the example of [1, -1] and [1, 1], the receiver may struggle to distinguish between transmitted signals. As a result, it becomes difficult to extract the original information accurately. In CDMA systems, achieving effective communication relies on selecting codes that do not interfere with each other, typically orthogonal ones, to ensure each piece of information can be distinctly deciphered by the receiving device.