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  • Moving from continuous- to discrete-time

    Sampling ideas Uniform, periodic sampling rate, e.g. CDs at 44.1KHz

    First we will need to consider periodic signals in order to appreciate how to interpret discrete-time signals as representative of continuous-time

    Fourier series and transforms

    Discrete-time Fourier transform (DTFT)

    Discrete Fourier transform (DFT) & the fast Fourier transform (FFT)

    Aliasing and signal reconstruction

    Discrete-time signals then discrete-time systems Matlab tools and digital signal processing

    Nothing on digital (finite bit-length) signals

    MAE143A Signals & Systems Winter 2016 1

  • Sampled signals discrete time

    For computer use we need to sample signals Most commonly this is periodic sampling every T seconds we take a sample of the continuous signal This yields a sequence

    A continuous-time signal is a function of time t A discrete-time signal is a sequence indexed by t

    T is the sampling period (sec)

    1/T is the sample rate (Hz)

    Often we also quantize the samples to yield a digital signal of a fixed number of bits representation

    This is a subject for the next course

    MAE143A Signals & Systems Winter 2016 2

    xn = x(nT )

  • Analog-to-digital conversion

    Signal conditioning Filtering (anti-aliasing), offset, ranging

    Track and hold circuit Enabled by the rising edge trigger Holds the signal at the latched value

    Analog-to-digital converter Converts held signal to 8-bit, 16-bit, 24-bit digital

    Computer coordinates sampling, storage, 1/T sec MAE143A Signals & Systems Winter 2016 3

    Electronic Signal

    Conditioner Real-world signal

    Track & hold Circuit ADC C

  • Ideal sampling of signals

    Continuous-time signal x(t) Continuous values of x, t

    Discrete-time signal xn Continuous x, integer n A sequence

    Sampled-data signal xs(t) Continuous x, t Modeled by a pulse train

    MAE143A Signals & Systems Winter 2016 4





    0 T 2T 3T 4T 5T 6T 7T -T -2T


    0 1 2 3 4 5 6 7 -1 -2 n

    xs(t) =1X

    n=1xn(t nT ) = x(t)


    n=1(t nT )

  • The Fourier transform (Roberts Chap 10)

    Generalization of the Fourier series to non-periodic function only one single, very long period

    X() is like the line spectrum with zero spacing between harmonic elements

    Still represents the signal as a sum of exponentials Requires absolute integrability of x(t)- finite energy

    Inverse Fourier transform

    MAE143A Signals & Systems Winter 2016 5

    T ! 1

    X() =

    Z 1

    1x(t)ejt dt

    x(t) =1


    Z 1

    1X(!)ej!t d!

  • The Fourier transform - example

    Two-sided decaying exponential

    MAE143A Signals & Systems Winter 2016 6

    X(!) =

    Z 1


    |t|5 ej!t dt


    Z 0


    t5j!t dt+

    Z 1


    t5j!t dt


    j! 15+


    j! + 15

    |X(!)| =25

    !2 + 125

    x(t) = e|t|5, t 2 (1,1)

  • The Fourier and Laplace transforms

    For a one-sided signal x(t)with Laplace transform

    Note that this is a signal property It mimics the Laplace transform and frequency response relationship for causal systems

    The interest in Fourier transforms is in the infinite-time response to periodic inputs

    Special Fourier transforms

    MAE143A Signals & Systems Winter 2016 7

    X(!) = X (s)|s=j!

    X (s)

    Y (!) = H(j!)U(!), H(!) = F(h(t))

    F [(t)] = 1, F [cos!0t] = [(! !0) + (! + !0)]F [1] = 2(!), F [sin!0t] = j[(! !0) + (! + !0)]

  • Bandlimited signals

    Signals, x(t), whose Fourier transform, X(), is zero for frequencies outside a value, the bandwidth, B, i.e.

    is referred to as bandlimited

    A periodic signal with line spectrum zero outside the bandwidth B is also referred to as bandlimited

    Bandlimited implies a level of smoothness

    Fourier series: finite power, periodic

    Fourier transform: finite energy, one-shot

    MAE143A Signals & Systems Winter 2016 8

    X(!) = 0 for {! > B} [ {! < B}

  • Fourier transform of a sampled signal

    The spectrum of a perfectly sampled signal xs(t) is a repeated and scaled version of X()

    Bandlimited signal bandwidth Bs/2

    Expect trouble in reconstruction of x(t) from xs(t)

    MAE143A Signals & Systems Winter 2016 9



    s 2s-s 0



    s 2s-s 0

  • Reconstructing sampled bandlimited signals

    Reconstruction of the original signal from samples

    We can reconstruct using an ideal lowpass filter This only works if

    is called the Nyquist sampling frequency

    MAE143A Signals & Systems Winter 2016 10


    s 2s-s 0B -B


    Lowpass Filter Bandwidth B

    !s 2B

    A signal of bandwidth B can be reconstructed completely and exactly from the sampled signal if the sampling frequency s is greater than 2B

    Nyquists Sampling Theorem

    !s = 2B

  • Sampling and reconstruction

    If the continuous-time signal is bandlimited to B and we sample faster than the Nyquist frequency

    Then we can exactly reconstruct the original signal from the sample sequence

    Conversely: if we sample at or more slowly than the Nyquist frequency then we cannot reconstruct the original signal from the samples alone

    We might have other information that helps the reconstruction. Without such extra information we suffer from Aliasing some frequencies masquerading as others

    MAE143A Signals & Systems Winter 2016 11

  • Aliasing

    Sampling rate is less than the Nyquist rate Spectra overlap in sampled signal Impossible to disentangle

    Signals from above s/2 masquerade as signals of a lower frequency

    This is what happens to wagon wheels in western movies

    fs=30Hz but the passing rate of the spokes is greater than this

    Dopplegnger and impersonation

    Alias: a false or assumed identity

    Samuel Langhorne Clemens alias Mark Twain

    5.5KHz sampled at 10KHz a.k.a. 4.5KHz

    MAE143A Signals & Systems Winter 2016 12

  • Aliasing II

    MAE143A Signals & Systems Winter 2016 13

  • Dealing with aliasing

    Anti-aliasing filters Before we sample a signal x(t) at fs we lowpass filter it through a filter of bandwidth Bfs/2

    This is straightforward but not simple 1. An ideal lowpass filter has non-timelimited

    non-causal response 2. A timelimited signal has a Fourier transform of

    unbounded support in frequency 3. We can only approximately lowpass filter

    1. We need to worry about roll-off near fs/2 2. We have to back off from fs/2 3. Fast roll-off is accompanied by passband ripple

    You cannot remove aliasing after sampling

    MAE143A Signals & Systems Winter 2016 14

  • Aliasing in practice

    Any data-logging equipment worth a dime has anti-aliasing filtering built in

    It is switched in at sample-rate selection time and is handled by the equipment software

    DSpace, LabView, HP analyzers Often this (and timing questions) limits the available sampling rates High-performance data acquisition is an art The data quality near Nyquist is dodgy

    So why do we need to know about aliasing? Because of the mistakes you can/might/will make!!

    MAE143A Signals & Systems Winter 2016 15

  • Downsampling scenario

    An email arrives containing two columns of data sampled at 44.1KHz

    You look at it and figure that it is oversampled and that most of the interesting information is in the range less than 10KHz in the signal

    This would be captured by data sampled at 22.5KHz

    So you toss out every second sample Now it is the same data sampled at half the rate

    Let us try this out with a former Hilltop High student

    MAE143A Signals & Systems Winter 2016 16

  • Downsampling

    MAE143A Signals & Systems Winter 2016 17

    Original data at 44.1KHz

    Downsampled data at 22.05KHz

    Decimated data at 22.05 KHz

    Difference at 22.05 KHz Between downsample and decimate

  • Decimation instead of downsampling

    The mistake earlier was that the reconstructed content of the 44.1KHz data contains signals in the range 0-20KHz roughly

    Downsampling to 22KHz and then reconstructing should yield signals in the 0-10KHz range

    The 10-20KHz material has been aliased into 0-10KHz

    Before downsampling we should filter the data with a lowpass digital filter of bandwidth ~10KHz

    This is called decimation

    MAE143A Signals & Systems Winter 2016 18

  • Matlabs decimate

    MAE143A Signals & Systems Winter 2016 19

    >> help decimate decimate Resample data at a lower rate after lowpass filtering. Y = decimate(X,R) resamples the sequence in vector X at 1/R times the original sample rate. The resulting resampled vector Y is R times shorter, i.e., LENGTH(Y) = CEIL(LENGTH(X)/R). By default, decimate filters the data with an 8th order Chebyshev Type I lowpass filter with cutoff frequency .8*(Fs/2)/R, before resampling. Y = decimate(X,R,N) uses an N'th order Chebyshev filter. For N greater than 13, decimate will produce a warning regarding the unreliability of the results. See NOTE below. Y = decimate(X,R,'FIR') uses a 30th order FIR filter generated by FIR1(30,1/R) to filter the data. Y = decimate(X,R,N,'FIR') uses an Nth FIR filter. Note: For better results when R is lar

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