# Outline. Discrete time signals. Impulse sampling z-transform Frequency response Stability INF4420. Jørgen Andreas Michaelsen Spring / 37 2 / 37

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1 INF4420 Discrete time signals Jørgen Andreas Michaelsen Spring / 37 Outline Impulse sampling z-transform Frequency response Stability Spring 2013 Discrete time signals 2 2 / 37

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 2013 Discrete time signals 3 3 / 37 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 2013 Discrete time signals 4 4 / 37

3 Introduction Spring 2013 Discrete time signals 5 5 / 37 Introduction Sample a continuous time input signal at uniformely spaced time points. Output is a discrete sequence of values (in theory). Spring 2013 Discrete time signals 6 6 / 37

4 Introduction Spring 2013 Discrete time signals 7 7 / 37 Sampling Laplace transform: Input signal Fourier transform: Spring 2013 Discrete time signals 8 8 / 37

5 Sampling Spring 2013 Discrete time signals 9 9 / 37 Sampling Impulse sampling Choose τ infinitely narrow Choose the gain k = 1 τ. The area of the pulse at nt is equal to the instantaneous value of the input at nt, f(nt). The signal is still defined for all time, so we can use the Laplace transform for analysis. Spring 2013 Discrete time signals / 37

6 Sampling Modelling the sampled output, f t We will model the sampled output in the time domain Then find an equivalent representation in the Laplace domain We will model each pulse independently and the whole signal by summing all pulses Spring 2013 Discrete time signals / 37 Sampling Modeling a single pulse using step functions Step function: u t 1, t 0 0, t < 0 Single pulse: f nt u t nt u t nt τ Spring 2013 Discrete time signals / 37

7 Sampling Sum all the pulses to get the sampled signal: f t = k f nt u t nt u t nt τ Spring 2013 Discrete time signals / 37 Sampling Transforming the time domain model to the Laplace domain Relevant Laplace transforms f(t) F(s) u(t) s 1 f t a u t a, a 0 e as F(s) Spring 2013 Discrete time signals / 37

8 Sampling Time domain model from before f t = k f nt u t nt u t nt τ Laplace domain F s = k f nt e snt s k 1 e sτ = s e s nt+τ s f nt e snt Spring 2013 Discrete time signals / 37 Sampling and the z-transform F s = k 1 e sτ s f nt e snt Impulse sampling: k = 1 τ, τ 0 (ex 1 + x) F s f nt e snt f nt z n Spring 2013 Discrete time signals / 37

9 The z-transform z e st X z x nt z n = x n z n Delay by k samples, z k X(z) Convolution in time multiplication in z Spring 2013 Discrete time signals / 37 Frequency response Use the Laplace domain description of the sampled signal As before, substitute s = jω X jω = x nt e jωnt n= X(jω) is the Fourier transform of the impulse sampled input signal, x(t). e jx is cyclic, e jx = cos x + j sin x Spring 2013 Discrete time signals / 37

10 Frequency response We go from the z-transform to the frequency response by substituting z = e jωt As we sweep ω we trace out the unit circle Spring 2013 Discrete time signals / 37 Frequency response Rewriting to use frequency (Hz), rather than radian frequency X f = x nt e j2πfnt n= Because e jx is cyclic, f 1 = k f s + f 1, where f 1 is an arbitrary frequency, k is any integer and f s is the sampling frequency (T 1 ). Spring 2013 Discrete time signals / 37

11 Frequency response The frequency spectrum repeats. We can only uniquely represent frequencies from DC to f s 2 (the Nyquist frequency). Important practical consequence: We must band limit the signal before sampling to avoid aliasing. A non-linear distortion. Spring 2013 Discrete time signals / 37 Frequency response Spring 2013 Discrete time signals / 37

12 Frequency response If the signal contains frequencies beyond f s 2, sampling results in in aliasing. Images of the signal interfere. Spring 2013 Discrete time signals / 37 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 2013 Discrete time signals / 37

13 Downsampling Keep every n-th sample. Downsample too much: Aliasing Spring 2013 Discrete time signals / 37 Upsampling Insert n zero valued samples between each original sample, and low-pass filter. Requires gain to maintain the signal level. Spring 2013 Discrete time signals / 37

14 Discrete time filters Analog filters use integrators, s 1, as building blocks to implement filter functions. Discrete time filters use delay, z 1. Example: Time domain: y n + 1 = bx n + ay[n] z-domain: zy z = bx z + ay(z) H z Y z X z = b z a Spring 2013 Discrete time signals / 37 Discrete time filters Frequency response, z = e jω (ω normalized to the sampling frequency, really z = e jωt ) H e jω = b e jω a DC is z = e j0 = 1. The sampling frequency is z = e j2π = 1 (also). Sufficient to evaluate the frequency response from 0 to π due to symmetry (for real signals). Spring 2013 Discrete time signals / 37

15 Stability y n + 1 = bx n + ay[n] If a > 1, the output grows without bounds. Not stable. H z = b z a In a stable system, all poles are inside the unit circle a = 1: Discrete time integrator a = 1: Oscillator Spring 2013 Discrete time signals / 37 IIR filters y n + 1 = bx n + ay[n] is an infinite impulse response filter. Single impulse input (x 0 = 1, 0 otherwise) results in an output that decays towards zero, but (in theory) never reaches zero. If we try to characterize the filter by its impulse response, we need an infinite number of outputs to characterize it. Spring 2013 Discrete time signals / 37

16 FIR filters y n = 1 3 x n + x n 1 + x n 2 Is a FIR (finite impulse response filter). H z = i=0 z i FIR filters are inherently stable but require higher order (more delay elements) than IIR. Spring 2013 Discrete time signals / 37 Bilinear transform Mapping between continuous and discrete time Design the filter as a continuous time transfer function and map it to the z-domain s = z 1 1+s, conversely, z = z+1 1 s s = 0 maps to z = 1 (DC) s = maps to z = 1 First order approximation f s 2 Spring 2013 Discrete time signals / 37

17 Sample and hold We modeled impulse sampling by letting τ 0. For the sample and hold, we use the same model, but let τ T. Use this to find the transfer function of SH Spring 2013 Discrete time signals / 37 Sample and hold F (s) = k 1 e sτ s f nt e snt Impulse sampling 1 Sample and hold: F k 1 e st (s) = s The pulse lasts for the full sampling period, T f nt e snt Spring 2013 Discrete time signals / 37

18 Sample and hold The sample and hold shapes spectrum H SH s 1 e st s Frequency (magnitude) response of the SH H SH jω = T sin ωt 2 ωt 2 Spring 2013 Discrete time signals / 37 Sample and hold Sampled signal spectrum Sample and hold sinc response, sin x x Spring 2013 Discrete time signals / 37

19 References Gregorian and Temes, Analog MOS Integrated Circuits for Signal Processing, Wiley, 1986 Spring 2013 Discrete time signals / 37

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