Chapter 7
The chapter discusses pipelining, retiming and lookahead techniques for transforming digital designs to meet the desired objectives. Broadly, signal processing systems can be classified as feedforward or feedback systems. In feedforward systems the data flows from input to output and no value in the system is fed back in a recursive loop. Finite impulse response (FIR) filters are feedforward systems and are fundamental to signal processing. Most of the signal processing algorithms such as fast Fourier transform (FFT) and discrete cosine transform (DCT) are feedforward. The timing can be improved by simply adding multiple stages of pipelining in the hardware design.
Recursive systems such as infinite impulse response (IIR) filters are also widely used in DSP. The feedback recursive algorithms are used for timing, symbol and frequency recovery in digital communication receivers. In speech processing and signal modeling, autoregressive moving average (ARMA) and autoregressive (AR) processes involve IIR systems. These systems are characterized by a difference equation. To compute an output sample, the equation directly or indirectly involves previous values of the output samples along with the current and previous values of input samples. As the previous output samples are required in the computation, adding pipeline registers to improve timing is not directly available to the designer. The chapter discusses cutset retiming and node transfer theorem techniques for systematically adding pipelining registers in feedforward systems. These techniques are explained with examples. This chapter also discusses techniques that help in meeting timings in implementing fully dedicated architectures (FDAs) for feedback systems.
The chapter also defines some of the terms relating to digital design of feedback systems. Iteration, the iteration period, loop, loop bound and iteration period bound are explained with examples. While designing a feedback system, the designer applies mathematical transformations to bring the critical path of the design equal to the iteration period bound (IPB), which is defined as the loop bound of the critical loop of the system. The chapter then discusses the node transfer theorem and cutset retiming techniques in reference to feedback systems. The techniques help the designer to systematically move the registers to retime the design in such a way that the transformed design has reduced critical path and ideally it is equal to the IPB.
Feedback loops pose a great challenge. It is not trivial to add pipelining and extract parallelism in these systems. The IPB limits any potential improvement in the design. The chapter highlights that if the IPB does not satisfy the sampling rate requirement even after using the fastest computational units in the design, the designer should then resort to lookahead transformation. This transformation actually reduces the IPB at the cost of additional logic. Lookahead transformation, by replacing the previous value of the output by its equivalent expression, adds additional registers in the loop. These additional delays reduce the IPB of the design. These delays then can be retimed to reduce the critical path delay. The chapter illustrates the effectiveness of the methodology by examples. The chapter also introduces the concept of ‘Cslow retiming’, where each register in the design is first replaced by C number of registers and these registers are then retimed for improved throughput. The design can then handle C independent streams of inputs.
The chapter also describes an application of the methodology in designing effective decimation and interpolation filters for the front end of a digital communication receiver, where the required sampling rate ismuch higher and any transformation that can help in reducing the IPB is very valuable.
IIR filters are traditionally not used in multirate decimation and interpolation applications. The recursive nature of IIR filters requires computing all intermediate values, so it cannot directly use polyphase decomposition. The chapter presents a novel methodology that makes an IIR filter use Nobel identities and effectively run at a slower clock rate in multirate decimation and interpolation applications.
