# Moving Average

In some application, we need to perform moving average on input data in our design.

In the moving average for each incoming sample, we need to perform an equation like

Starting from this equation we should perform the average computation for each input sample.

## Why we need an encoder

In the communication link, the noise, multipath effect and all the other effects that affect the communication can induce errors on the communication link. In this post, we already addressed the topic introducing the Reed-Solomon encoder.

Here we want to exploit another encoder widely used in communication link: Convolutional Encoder.

After a brief introduction to the Convolutional encoder, we will go to see the thing that interests us the most, I mean the VHDL implementation of a Convolutional Encoder.

## The errors occur in burst

When we establish a communication between two or more devices we are prone to errors.

In this post, we can see how to prevent errors using a coding technique like Reed-Solomon coding. There are a lot of coding techniques in the literature.

But the only data coding is not enough for a good prevention of communication error.

The encoders/decoders work fine on a “sparse” error condition. In a communication channel errors occur in bursts.

When you have a perturbation over a channel, it persists on the channel for a certain amount of time.

During the perturbation on the communication link, the number of bits involved will be proportional to the speed of the link: higher is the link speed, higher will be the consecutive number of bits that can be affected.

So, what could be a solution to this issue?

## FIR filter overview

When we implement an FIR filter in FPGA or ASIC the major cost in terms of area resources derives from the multiplier units required for coefficient multiplication. Depending on the frequency response of the filter, the number of coefficients (or filter taps) could become large.

As clear from an FIR filter architecture in Figure 1 the number of multiplier operations is equal to the number of taps, for a long filter the area required will be demanding.

## What is an RS-Encoder

When you want to transfer information from a source to a target, you want to be sure that such information is transferred without errors.

When you transfer an information through a communication channel, such information will be prone to errors.

In order to minimize or eliminate the number of errors during the information transmission, we can adopt some error correction strategies.

Such strategies are named FEC, i.e. Forward Error Correction. As you can imagine, nothing is for free. If you want to guarantee an error-free transmission you need to “pay” in terms of transmission bandwidth and transmitter/receiver complexity.

Any error correction strategy needs to transmit much more information than the minimum required, in order to allow the receiver to recognize if the received information is correct and, in case of error, provide to correct the errors.

Without entering in the error correction field theory, very complex and wide argument, here we want to address one possible error correction algorithm.

There are two main approaches to implement Error- Correction Coding:

• Block Coding: the symbol stream is divided into a block and coded.
• Convolution Coding: convolution operation is applied to the symbol stream.

## Why Galois field

The Galois fields are mainly used in cryptography and error correction algorithm.

If you never deal with Galois field, at the beginning the topic could seem very hard to understand. In this post we want to address the galois field theory from the practical application point of view.

We will review:

• the Galois arithmetic notation, just to understand how to interpret the equation
• add/sum operation in Galois field
• multiplier in Galois field

the third point maybe is the most difficult to understand.

It might put noses out of joint after reading this post because I will not be very rigorous in the representation of the theory of the Galois fields. This post wishes to give you a practical implementation starting point for your VHDL design using Galois arithmetic.

I put a lot of references at the bottom of this post that will help you to go deeper in the Galois field theory if you are really interested in it.

## FPGA and Processor

The modern FPGAs implement microprocessor internally either as soft-processor or hard-processor. The soft-processor is intended as microprocessor implemented into the FPGA starting from a VHDL/Verilog code.

So, in this case, the processor is synthesized using the current FPGA technology (and layout tool).

Soft-processors have the advantages to be portable into a different technology. LEON processor is a soft-processor example that is realized as a stand-alone chip and can be integrated into a VHDL design as IP.

The Hard-processor is intended as a dedicated FPGA silicon area that implements the processor.

As you can understand, it is dependent on the FPGA are you using. For example, Intel Cyclone V SoC family implements a dual-core processor ARM-Cortex A9.

Another example is the Xilinx ZYNQ that implement different processor families, ARM-Cortex A9, Arm Cortex-A53.

When we use a processor in FPGA (hard-processor or soft-processor) sometimes we need to interface such processor with our custom peripheral implemented in VHDL (or Verilog or other custom implementation). In this case, we could experience different issues in the processor – peripheral interfacing:

• A bus of different size
• Different interface signals

# What is a MUX?

When we implement a digital hardware architecture, we often need to select an input to our logic between several different inputs. This selection logic is called digital multiplexer or MUX.

We name it digital multiplexer, to distinguish it from an analog multiplexer. An analog multiplexer implements the same function as digital MUX selecting the source of a signal from different analog source instead of digital.

As clear in Figure1, a MUX can be visualized as an n-way virtual switch whose output can be connected to one of the different input sources. On the left side of the Figure1, you can see the typical MUX representation. The number near the input ports indicates the selector value used to route the selected input to the output port.

## What is an LFSR

A linear-feedback shift register (LFSR) is a shift register whose input bit is a linear function of its previous state. We can use this type of functions in many application such as counters, crypto, ber-meter, CRC generation, scrambling/descrambling algorithm, test application and so on

An LFSR of length N can generate 2^N-1 different states where the values look like pseudo-random values.

There are two different types of LFSR implementation the FIBONACCI and the GALOIS implementation as in Figure1. The LFSR implementations are equivalent.

If we are implementing the LFSR in hardware, the Galois implementation is much more efficient since use two input XOR function and the XOR function is implemented between two consecutive registers. Read More

## VHDL integer division should be really avoided?

In VHDL there are the math primitive subtraction, addiction, and multiplication that are generally available in the libraries provided by the FPGA or ASIC vendor.

For example, in this post, we saw how to implement a pipelined multiplier. The example shows the use of multiplication and addition primitives.

The division is a bit more complex case. Generally, the deployment of the division requires a much more complex logic circuit, and for this reason, we tend to avoid, where possible, the use of the division operator unless there are special cases.

If we have to divide by 2 or power of two, the implementation is simply shifting to the right of the number to divide.

In binary representation, shifting to the right of a position corresponds to a division by two, as in a decimal representation a shifting to the right corresponds to a division by 10.