As the title implies, I assume in this post that you are somewhat familiar with fixed-point arithmetic and IIR filters. It is common for the design of a quantizer to involve determining the proper balance between granular distortion and overload distortion.
For example, we can generate a direct form structure from the difference equation, or equivalently from the z transform. Then we will quantize the coefficients of these second-order sections and compare the frequency response of the obtained structure with that of the unquantized system.
Suppose we want to represent the number three. As there are conflicting formats, I opted for the Texas Instruments version Qm. As for the quantization noisea statistical model for the round-off error in multipliers is used to quantify the average impact of round-off noise in a non-recursive filter.
Then, the purpose here is to address the drawback — loss of precision — to prevent it from compromising the final result. To form the next term in the sum for n equals 1, we have the output of one delay plus the output from the 7th minus 1th delay.
And when little n is equal to capital N minus 1, little r is equal to 0. And now that is multiplied by h of 1 coefficient, which is h of 1, to form the next term in the sum. And we can change it back to n if we want to, which we do. And basically where we want to get to is to exploit the fact that these points are numerically equal to these, except that the index runs in the other direction.
Consider now that we have a microcontroller with 8-bit word length, that is, each register contains a bit for the sign and seven bits to store the number. First, if we multiply 0.
Before we go into details, let us recall how microcontrollers and DSPs — at least the majority of them — interpret binary numbers. Both the input signal and filter coefficients are quantized.
Here we have a sum from r equals 0 to N over 2 minus 1, some things involving h of r. The resulting quantization error is known as round-off error. And, obviously, each of these quantized coefficients will contribute a particular amount of error to the overall error.
We first take a look on the quantization of the filter coefficients, followed by the effects caused by a finite numerical resolution on the operations.
And a similar derivation would apply and has gone through in the text for capital N odd. So the first point then is that we can apply many of the ideas of the previous lecture to FIR systems. First, the choice for a fixed-point IIR filter provides us with a fair amount of information about what the designer has in mind.
This sum, of course, stays the same. So this is an argument, again, that stresses the fact that if the unit sample response has this symmetry, then the phase of the system is linear.
The reason for that being that the coefficients in the structure correspond to samples of the frequency response of the system.
And consequently, its Fourier transform is a real function of omega. And then likewise, the linear phase property says that as I go through the impulse response from both ends, I see identical values so that these values are identical.
However, in some quantizer designs, the concepts of granular error and overload error may not apply e.
Equation 2 has two important implications that will be discussed next. Now, this unit sample response is even. For instance, we can assume execution time is a major concern, which is reasonable for most real-time applications, and he is willing to trade some level of precision for speed.
Looking at this expression for n equal 0, no delay. Quantization error models[ edit ] In the typical case, the original signal is much larger than one least significant bit LSB.Fixed Point Effects in Digital Filters Cimarron Mittelsteadt David Hwang Finite-precision Problems Quantizers are nonlinear devices Characteristics may be significantly Digital Filters Coefficient Quantization Frequency Response.
ece/, digital filter structures and quantization effects 6–2 We will see that the performance of a digital implementation is affected substantially by the choice of. Quantization effects in digital filters can be divided into four main categories: quantization of system coefficients, errors due to analog-digital (A-D) conversion, errors due to roundoffs in the arithmetic, and a constraint on signal.
• Filter Quantization • Digital Filters Effects • Digital Filters Design • FIR Filters Design • FIR Windows Design - 1 • FIR Windows Design - 2 • Frequency Sample Method Quantization Effects in the Computation of DFT Posted on October 27, by Manish.
Digital filters is a great topic and I could dedicate the whole post to the advantages they have compared to their analog counterparts (despite some drawbacks). However, designing and implementing these fine pieces can be quite tricky, and if proper care is not taken the effects can be catastrophic.
This lecture covers direct form FIR filters, efficient implementation of FIR filters with linear phase, frequency sampling structure, and the effects of parameter-quantization in digital filter implementation.Download