INF4420 Discrete time signals

Dag T. Wisland Spring 2014 Outline

• Impulse sampling • z‐Transform • • Stability

Spring 2014 Discrete time signals 2 Introduction

• More practical to do processing on sampled signals in many cases • Sampled + quantized signals = digital • Inputs and outputs are not sampled • How does sampling affect the signals? • Tools for analyzing sampled signals and systems (“discrete ”, the z‐transform)

Spring 2014 Discrete time signals 3 Introduction

• We have already seen sample and hold circuits • We can also realize integrators, filters, etc. as sampled analog systems—switched capacitor techniques. Discrete time, continuous amplitude. • Digital processing is efficient and robust, usually preferred where applicable. Sampling also applies to digital.

Spring 2014 Discrete time signals 4 Introduction

Spring 2014 Discrete time signals 5 Introduction

• Sample a continuous time input signal at uniformely spaced time points. • Output is a discrete sequence of values (in theory).

Spring 2014 Discrete time signals 6 Introduction

Spring 2014 Discrete time signals 7 Sampling

Laplace transform: Input signal

Fourier transform:

Spring 2014 Discrete time signals 8 Sampling

Spring 2014 Discrete time signals 9 Sampling

Spring 2014 Discrete time signals 10 Sampling

Spring 2014 Discrete time signals 11 Sampling

Spring 2014 Discrete time signals 12 Sampling

Spring 2014 Discrete time signals 13 Sampling

Spring 2014 Discrete time signals 14 Sampling

Spring 2014 Discrete time signals 15 Spring 2014 Discrete time signals 16 Spring 2014 Discrete time signals 17 Frequency response

Spring 2014 Discrete time signals 18 Frequency response

Spring 2014 Discrete time signals 19 Frequency response

Spring 2014 Discrete time signals 20 Frequency response

Spring 2014 Discrete time signals 21 Frequency response

Spring 2014 Discrete time signals 22 Frequency response

Spring 2014 Discrete time signals 23 Sampling rate conversion

• Changing the sampling rate after sampling • We come back to this when discussing oversampled converters • = sampling faster than the would indicate • is increasing the sampling rate (number of samples per unit of time) • Downsampling is decreasing the sampling rate

Spring 2014 Discrete time signals 24 Downsampling

Keep every n‐th sample. Downsample too much:

Spring 2014 Discrete time signals 25 Upsampling

Insert n zero valued samples between each original sample, and low‐pass filter. Requires gain to maintain the signal level.

Spring 2014 Discrete time signals 26 Discrete time filters

Spring 2014 Discrete time signals 27 Discrete time filters

Spring 2014 Discrete time signals 28 Stability

Spring 2014 Discrete time signals 29 IIR filters

Spring 2014 Discrete time signals 30 FIR filters

Spring 2014 Discrete time signals 31 Bilinear transform

Spring 2014 Discrete time signals 32 Sample and hold

Spring 2014 Discrete time signals 33 Sample and hold

Spring 2014 Discrete time signals 34 Sample and hold

Spring 2014 Discrete time signals 35 Sample and hold

Sampled signal spectrum

Spring 2014 Discrete time signals 36 References

Gregorian and Temes, Analog MOS Integrated Circuits for , Wiley, 1986

Spring 2014 Discrete time signals 37