INF4420
Discrete time signals
Dag T. Wisland
Spring 2014
Outline
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Impulse sampling
z‐Transform
Frequency response
Stability
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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)
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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.
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Introduction
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Introduction
• Sample a continuous time input signal at uniformely spaced time points.
• Output is a discrete sequence of values (in theory).
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Introduction
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Sampling
Laplace transform:
Input signal
Fourier transform: Spring 2014
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Sampling
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Sampling
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Sampling
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Sampling
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Sampling
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Sampling
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Sampling
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Frequency response
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Frequency response
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Frequency response
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Frequency response
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Frequency response
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Frequency response
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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
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Downsampling
Keep every n‐th sample. Downsample too much: Aliasing
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Upsampling
Insert n zero valued samples between each original sample, and low‐pass filter. Requires gain to maintain the signal level.
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Discrete time filters
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Discrete time filters
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Stability
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IIR filters
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FIR filters
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Bilinear transform
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Sample and hold
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Sample and hold
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Sample and hold
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Sample and hold
Sampled signal spectrum
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References
Gregorian and Temes, Analog MOS Integrated Circuits for Signal Processing, Wiley, 1986
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