VI Quantisation

When the signal is converted to digital form, the precision is limited by the number of bits available.

The diagram shows an analogue signal which is then converted to a digital representation - in this case, with 8 bit precision.

The smoothly varying analogue signal can only be represented as a 'stepped' waveform due to the limited precision.

Sadly, the errors introduced by digitisation are both non linear and signal dependent.

Non linear means we cannot calculate their effects using normal maths.

Signal dependent means the errors are coherent and so cannot be reduced by simple means.

This is a common problem in DSP. The errors due to limited precision (ie word length) are non linear (hence incalculable) and signal dependent (hence coherent). Both are bad news, and mean that we cannot really calculate how a DSP algorithm will perform in limited precision - the only reliable way is to implement it, and test it against signals of the type expected. The non linearity can also lead to instability - particularly with IIR filters.

Introduction to DSP - filters

This is the fifth module of the BORES Signal Processing DSP course - Introduction to DSP. It covers the following subjects:

 

The word length of hardware used for DSP processing determines the available precision and dynamic range.

Precision

Precision measures the number of bits that can be used to represent each number. Precision determines the extent of errors caused by integer truncation or rounding - called quantisation errors.

Dynamic range

Dynamic range is the range of numbers that can be represented before overflow (or underflow). Dynamic range determines when signals need to be scaled - and since scaling takes processor cycles, this affects the speed of DSP operations. Scaling also causes errors due to integer truncation or rounding - called quantisation errors.

 

 

Uncertainty in the clock timing leads to errors in the sampled signal.

The diagram shows an analogue signal which is held on the rising edge of a clock signal. If the clock edge occurs at a different time than expected, the signal will be held at the wrong value. Sadly, the errors introduced by timing error are both non linear and signal dependent.

A real DSP system suffers from three sources of error due to limited word length in the measurement and processing of the signal:

These errors are often called 'quantisation error'. The effects of quantisation error are in fact both non linear and signal dependent. Non linear means we cannot calculate their effects using normal maths. Signal dependent means that even if we could calculate their effect, we would have to do so separately for every type of signal we expect. A simple way to get an idea of the effects of limited word length is to model each of the sources of quantisation error as if it were a source of random noise.

The model of quantisation as injections of random noise is helpful in gaining an idea of the effects. But it is not actually accurate, especially for systems with feedback like IIR filters.

The effect of quantisation error is often similar to an injection of random noise.

The diagram shows the spectrum calculated from a pure tone

The effect looks very like low level random noise. The signal to noise ratio is affected by the number of bits in the data format, and by whether the data is fixed point or floating point.

Fixed point DSP

Fixed point DSP hardware has lower dynamic range and precision than floating point devices. But it is preferred for some applications (notably, data compression is often better done by a fixed point DSP processor), and some fixed point devices (such as the Motorola DSP56301) may be faster than floating point devices. Fixed point processors are also cheaper, so are often preferred when designing a custom system which will be produced in some quantity. Our fixed point DSP products use the Analog Devices ADSP2181 16 bit processor which calculates at 33 MIPS (millions of instructions per second), and the Motorola DSP56002 and DSP56301 24 bit processors, which calculates at 33 MIPS and 66 MIPS respectively. The extended dynamic range and precision of the 24 bit processors means they are often preferred for applications such as high quality audio - whereas the 16 bit processor is cheaper and often used for speech or telecommunications applications.

Floating point DSP

Floating point DSP hardware has many advantages for DSP. It simplifies programming, and makes it easier to use high level languages. It also offers greater dynamic range and precision than fixed point devices. But is is worth noting that floating point is not best for all applications (notably, data compression is often better done by a fixed point DSP processor). Our floating point DSP products use the Lucent DSP32C processor which calculates at 25 Mflop (millions of floating point operations per second) or 40 Mflops, depending on the processor clock speed (50 MHz or 80 MHz), or the Texas TMS320C32 processor which calculates at 33 Mflop.