Bandwidth
- The highest frequency component of an analog waveform that a system would have to be
capable of processing without distorting the signal. See Sampling
and Time Domain.
Coefficient
- A coefficient is simply a factor which is used in a multiplication operation. The most
common usage of this term is with respect to multiplication performed by filters. You will
hear of constant coefficients for a filter whose characteristics never change, or adaptive
coefficients for filters whose characteristics change over time. Coefficients are
calculated for the desired frequency response of the filter using "magic".
Correlation
- This is the process of scanning data for a certain pattern. In video applications the
goal could be to isolate a specific pattern, or to just look for a boundary. The types of
correlation being performed can be multi-dimensional, hence the use of 1-D, 2-D or 3-D
correlators.
Data Width
- When data is input to a digital system it is represented as a binary number of a certain
bit width. Generally, more bits means better accuracy in the digital representation. Two
other terms are used in conjunction with the data width: precision and dynamic
range. For a given data width, the range of the highest value that can be represented
down to the lowest value that can be represented is called the dynamic range (usually
quoted in dB) with a certain resolution, called the precision.
The reason that this matters is because multiplication of numbers can cause word growth.
As an example, we know that multiplying two 8-bit numbers can result in data which is up
to 16 bits wide if the multiplicands are greater than 1. If the numbers being multiplied
are scaled so that they are less than 1, then the result will also always be less than 1,
and can be represented using 8-bits.
DCT
- Discrete Cosine Transform. A common form of mathematical algorithm used in
compression of audio and video data. Examples of standards that use DCT are:
- Dolby AC2 & AC3 - 1-D DCT (and 1-D Discrete Sine Transform)
- JPEG (still images) - 2-D DCT spatial compression
- MPEG1 & MPEG2 - 2-D DCT plus motion compensation.
Decimation
- Decimation is the opposite of interpolation. Sometimes the
effective sampling rate of a signal must be increased or decreased. This may be required
when data has to be transferred from one type of system to another. The process of
increasing the sampling rate is called interpolation, while the process of decreasing the
sampling rate is called decimation. The most important consideration is that the process
of interpolation or decimation not destroy the signal's integrity, i.e., the signal still
be a reasonably accurate representation of its analog equivalent after the process of
interpolation or decimation.
Note: Decimation also has a specialized meaning with respect to FFTs.
FIR
- Finite Impulse Response. The impulse response of a system is the name given to what
happens at the system's output when you input a single pulse. In the case of an FIR
filter, the output eventually settles down (i.e., is finite, as opposed to infinite--see
IIR). FIR filters can also be referred to as moving average filters. Other terms that you
are likely to hear when people talk about these are:
- Order - The complexity of the filter's frequency response.
- Taps - The number of time delayed inputs and coefficients in the filter's
mathematical definition.
- Symmetrical - If the coefficients of the filter's mathematical definition are
symmetrical about the center then the filter is said to be symmetrical. A symmetrical
filter's transfer function would look something like:
- C0.X0 + C1.X1 + C2.X2 + C3.X3 + C4.X4 + C3.X5 + C2.X6 + C1.X7 +C0.X8
In the case of a non-symmetrical filter the transfer function would be:
- C0.X0 + C1.X1 + C2.X2 + C3.X3 + C4.X4 + C5.X5 + C6.X6 + C7.X7 +C8.X8
Note that in each of the above examples the filter, would be 3rd Order and have 9-taps.
- Adaptive - A filter is made up of many multiplications of time delayed inputs by
coefficients. If the coefficients are changed at any time, the filter is said to be
adaptive. The implications of this for the design of the filter is that the multipliers
usually become more complicated than the case where the coefficients are constant.
These terms also apply to IIR filters.
FFT
- Fast Fourier Transform. This is an accelerated algorithm for performing a Discrete
Fourier Transform (DFT). An FFT is used to calculate a waveform's frequency spectrum (see time domain and frequency domain). It is often
easier to analyze a waveform in the frequency domain than it is in the time domain (e.g.,
radar reflections, speech, data compression). Also referred to as DFT (Discrete Fourier
Transform).
In an n-point FFT, the number of "points" is used to refer to the amount
of input data (the data set) on which you are going to perform the FFT calculation. The
more points, the bigger the calculation.
Note: Xilinx devices can do 1024-point FFTs.
The radix of an FFT is another term you may come across. A radix 2 FFT supports
data set sizes that are powers of 2. A radix 4 FFT supports data set sizes that are powers
of 4. Radix 4 FFTs are less flexible, but require less multiplication to be performed and
are therefore faster.
The term decimation is also used in a specific context with respect
to FFTs. The process of performing the FFT function scrambles the data that is generated.
If the time domain input is regularly ordered, the frequency domain output is scrambled (decimation
in frequency). If the time domain input is scrambled appropriately, the frequency
domain output is properly ordered (decimation in time).
The term re-ordering is used to refer to the unscrambling of data.
Frequency Domain
- See Time Domain
iFFT
- inverse Fast Fourier Transform. This is a mathematical function which generates the time
domain representation (a waveform) from a frequency spectrum (see Time
Domain, Frequency Domain and FFT).
IIR
- Infinite Impulse Response. The impulse response of a system is the name given to what
happens at the system's output when you input a single pulse. In the case of an IIR filter
the output never settles down (i.e., it varies infinitely - as opposed to being finite,
see FIR). This is because these filters include feedback from the
output. While these filters have more capabilities than FIR filters they are prone to
instability and their design requires more care. Certain types of IIR filters are also
referred to by special names such as a BiQuad filter, which is a second order IIR filter.
Interpolation
- See Decimation
Nyquist
- See Sampling
Parallel Arithmetic
- See PDA
PDA
- Parallel Distributed Arithmetic. This is a way of implementing multiplication using a
fully parallel process. While costly in terms of silicon area, it yields the fastest
results.
Phase Response
- You hear a lot of talk about the frequency response of a system (i.e., how it mashes the
frequencies that are input to it). There is another aspect of the system's effect on the
input signal which is important in certain applications, namely the phase response. The
phase response of a system is how the system mashes the phase of a signal that passes
through it.
Precision
- The term precision can be used in relation to multiplication. When two numbers are
multiplied together the result is typically a larger number than either of the
multiplicands. For example, multiplying two 8-bit quantities can result in a number which
is as large as 16-bits. In some cases, the full data width is required (i.e., all 16-bits,
in the above example). This is called maximum precision. If only the most significant
8-bits would be required, the term 8-bit precision would be used.
Sampling
- The process of generating a digital equivalent for an analog waveform involves measuring
the analog voltage at various instants in time and concatenating these as a series of
digital numbers. This process is referred to as sampling and the rate at which the
measurements are made is called the sample rate. There is a rule that says in order to
avoid loss of quality of a signal the sampling rate must be greater than or equal to twice
the highest frequency contained in the analog signal. For example, female human speech
typical contains frequencies of up to 16KHz maximum (i.e., a 16KHz bandwidth) and so the
minimum sampling rate that could be used without signal degradation would be 32KHz. This
is also referred to as the Nyquist frequency.
SDA
- Serial Distributed Arithmetic. This is a way of implementing multiplication using a
serial process. The advantage of this technique is that the logic required for such an
implementation is very small. The corresponding disadvantage is that to perform one
multiplication requires several clock cycles, and so the data throughput rate is low.
Serial Arithmetic
- See SDA
Time Domain
- When looking at a signal we tend to naturally think of the signal as varying in time.
The easiest signal to think of is a sine wave, which most of us have seen on an
oscilloscope before. A sine wave has a single frequency and a known amplitude.
There is nothing wrong with thinking about a signal in this way, but sometimes if the
signal is particularly complex, there may be a better way of looking at it. One way to
consider looking at the signal is from a frequency point of view.
When doing signal analysis, the mathematics is often easier if you work with the frequency
representation of a signal. As a result, a complex waveform is often expressed in terms of
its frequency components when it is manipulated (see FFT). This is
referred to as working in the Frequency Domain.
|
