(DSP) 1 - The Representations for Signal and Digital Signal Processing System

The Representations for Signal and Digital Signal Processing System

The Fundamental Theory of Digital Signal Processing
School of Software, Shandong University

 

1.1 Characterization and classification of signals

• Continuous-time signal : \(u(t)\)

  \(t\), second ---- continuous independent variable

  • with continuous-valued amplitudes : analog signal
  • with discrete-valued amplitudes : quantized boxcar signal

• Discrete-time signal : \(\{v(nT)\}=\{v[n]\}\)

  \(n\) ---- discrete independent variable

  \(T\), second ---- Sampling interval or sampling period

  \(f_s=\frac{1}{T}\), Hz ---- Sampling frequency

  \(v[n]\), each member ---- sample

  • with continuous-valued amplitudes : sample-data signal
  • with discrete-valued amplitudes : digital signal

​ Page 3, Figure 1.1

​ In this course, digital signal is treated same as sample-data signal.

​ Presentation of digital signal

• Function

  ex. \(f[n]=\pi n^2\)

• Sequence

  ex. \(x[n]=\{10, -7, 10, 9, 11\},~-2\leq n \leq 2\)

 

1.2 Some basic digital signals (sequence)

• Unit Sample Sequence

\[\delta[n]= \left\{ \begin{array}{lr} 1, & n=0 \\ 0, & n\neq 0 \end{array}\right. \nonumber \]

​ the unit sample sequence shifted by \(k\) samples (ex. delay \(k\) samples)

\[\delta[n-k]= \left\{ \begin{array}{lr} 1, & n=k \\ 0, & n\neq k \end{array}\right. \nonumber \]

​ Representation of an Arbitrary Sequence

​ ex. \(x[n]=\{10, -7, 10, 9, 11\},~-2\leq n \leq 2\)

\[\begin{align} x[n] &= \sum\limits_{k=-2}^{2}\delta[n-k]x[k] \nonumber \\ &= 10\delta[n+2] -7\delta[n+1] + 10\delta[n] + 9\delta[n-1] + 11\delta[n-2] \nonumber \end{align} \]

• Unit Step Sequence

\[\mu[n]= \left\{ \begin{array}{lr} 1, & n\geq0 \\ 0, & n < 0 \end{array}\right. \nonumber \]

​ the unit step sequence shifted by \(k\) samples (ex. delay \(k\) samples)

\[\mu[n-k]= \left\{ \begin{array}{lr} 1, & n\geq k \\ 0, & n < k \end{array}\right. \nonumber \]

​ the relation between \(\delta[n]\) and \(\mu[n]\)

\[\begin{align} \mu[n] &= \sum\limits_{m=0}^\infin \delta[n-m]=\sum\limits_{k=-\infin}^n \delta[k] \\ \delta[n] &= \mu[n] - \mu[n-1] \end{align} \]

• Exponential Sequence

​ Decay & Growing : \(x[n]=\alpha^n\)

​ Euler's formula is the fundamental of fourier transform.

\[e^{j\theta} = \cos\theta + j\sin\theta \]

• Sinusoidal digital

\[A\sin(\Omega t + \phi) ~\Rightarrow~ \text{sampling } t=nT ~\Rightarrow~ A\sin(\omega_0 n+\phi) \]

​ \(A\) ---- amplitude \(\phi\) ---- phase \(T\) ---- sampling

​ \(\Omega\), \(rad/s\) ---- angular frequency \(\omega_0\), \(rad\) ---- normalized angular frequency

\[\omega_0 = \Omega T \]

​ Suppose that sampled signal is periodic, denote \(N\) as period of sinusoidal frequency,

\[\omega_0(n + N) = \omega_0 n + 2k\pi ~~\Rightarrow ~~\omega_0 N = 2k\pi ~~\Rightarrow~~ N=\frac{2k\pi}{\omega_0} \]

​ which turns out that \(\omega_0\) should be the form of \(k_0\pi,~k_0 \in Z^*\).

​ In this situation, normally, \(-\pi \leq \omega_0 \leq \pi\).

 

1.3 The presentation of general digital signal

• Enumeration form

\[x[n] = \{...,x[-1],x[0],x[1],...\} \]

• Convolutional form (time shift property)

\[x[n]=\sum\limits_{m=-\infin}^\infin \delta[n-m]x[m] = \delta[n] \otimes x[n] \]

• Using Euler's formula

\[\begin{align} \cos(\omega_0 n + \phi) &+ j\sin(\omega_0 n + \phi) = e^{j(\omega_0 n + \phi)} \\ \cos(\omega_0 n + \phi) &= \Re(e^{j(\omega_0 n + \phi)}) = \frac{e^{j(\omega_0 n + \phi)} + e^{-j(\omega_0 n + \phi)}}{\small 2} \\ \sin(\omega_0 n + \phi) &= \Im(e^{j(\omega_0 n + \phi)}) = \frac{e^{j(\omega_0 n + \phi)} - e^{-j(\omega_0 n + \phi)}}{\small 2j} \end{align} \]

 

1.4 The presentation of DSP system

• Basic operations

  • Multiplication (Product)

    • Modulation
    • Windowing

  • Scale

  • Addition & Subtraction

    ex. reduce noise

  • Time-shift (Delay)

    ex. it could be cascaded

  • Time-reversal

  • Pick-off

  Page 39, Figure 2.5

• General representation of DSP system

  • Differential form

    \(\sum a_q \cdot y[n-q] = \sum b_p \cdot x[n-p]\)

    • No-recursive form

      ex. \(y[n] = \large\frac{1}{1+m} \normalsize\sum\limits_{k=0}^m x[n-k]\)

    • Recursive form

      ex. \(y[n] = y[n-1] + \large\frac{1}{1+m} \left[ \normalsize x[n] - x[n-1-m] \large\right]\)

  • Convolution form

    \(y[n] = \sum b_p \cdot x[n-k] ~\to~ \sum\limits_{k=-\infin}^\infin h[k]x[n-k] ~\to~ \sum\limits_{k=-\infin}^\infin x[k]h[n-k]\)

    ex. \(y[n] = \large\frac{1}{1+m} \normalsize\sum\limits_{k=0}^m x[n-k]\)

 

1.5 Some popular discrete-time system

• Up-sampling

\[y[n] = \left\{ \begin{array}{cl} x[\frac{n}{L}], & n=0, \pm L, \pm 2L, \dots \\ 0, & \text{otherwise} \end{array} \right. \]

• Down-sampling

\[y[n] = \left\{ \begin{array}{cl} x[nL], & n=0, \pm L, \pm 2L, \dots \\ 0, & \text{otherwise} \end{array} \right. \]

• Accumulator

  Three forms of accumulator,

  \[\begin{array}{l} y[n] = \sum\limits_{l=-\infin}^n x[l] \\ y[n] = \sum\limits_{k=0}^{\infin} x[n-k] \\ y[n] = y[-1] + \sum\limits_{l=0}^n x[l] \end{array} \]

  In the last equation, \(y[-1]\) is the notation of \(\sum\limits_{l=-\infin}^{-1} x[l]\) , which is known previously.

• M-point moving-average system

\[y[n] = \frac{1}{M}\sum\limits_{k=0}^{M-1} x[n-k] \]

​ It's used for noise reducing.