INF4420 Discrete time signals
Dag T. Wisland Spring 2014 Outline
• Impulse sampling • z‐Transform • Frequency response • 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 Laplace transform”, 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 • Oversampling = sampling faster than the Nyquist frequency would indicate • Upsampling 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: Aliasing
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 Signal Processing, Wiley, 1986
Spring 2014 Discrete time signals 37