• Approximate an Ideal Lowpass filter

  • Pick cutoff frequency

  • Compute ideal impulse response

  • Truncate it to a finite-support

  • Defines an FIR filter

I thought why we cannot just use 1 and 0 as a step function in the frequency domain. But this is the step function in Frequency domain.

I thought why we cannot just use 1 and 0 as a step function in the frequency domain. But this is the step function in Frequency domain.

• Modulation Theorem

  • Recall DTFT{(x*y)}[n] = X(e^{jw})Y(e^{jw}) [Convolution Theorem]

  • DTFT{x[n] y[n]} = (X * Y)(e^{jw})

  • Recall,
    
    
    
    Then in L_2([-\pi, \pi]),
    

• z-transform

  • LTI system
    
    
    
    

  • The z-transform
    
    
    
    
    
    Consider an LTI system with impulse response h[n]
    
    
    
    H(z) is the transfer function of the system.
    
    
    
    
    

  • Example: Leaky Integrator

    • y[n] = (1-\lambda) x[n] + \lambda y[n-1] implies h[n] = (1-\lambda) lambda^n u[n]

    • transfer function from impulse response

• Existence and region of convergence (ROC)

  • ROC is defined by the absolute convergence of the power series:
    
    
    
    

  • Observation 1: for finite-support signals (FIR), the z-transform converges everywhere (except in 0 and/or \infty)

  • Observation 2: The ROC has circular symmetry -> z = ae^{j \theta}

  • Observation 3: for causal sequences, the ROC extends from a circle to infity: assuming z_0 \in ROC and |z_1| > |z_0|.

    • This means, for causal sequence, the ROC converges everywhere in the outside of the circle with radius |z_0|.

    • LTI system:
      
      
      
      where z = p_n are called Poles and z = z_n are called zeros.

  • BIBO stability <=> \sum_{n=-\infty}^{\infty} |h[n]| < \infty

  • 1 \in ROC <=> \sum_{n=-\infty}^{\infty} |h[n]| < \infty because H(z) converges absolutely in z= 1

    • This implies a causal system is stable iff ROC includes the unit circle.

  • For example,
    
    
    
    
    This is unstable because the largest ROC is 2.

• Resonator: a narrow bandpass filter

  • DTMF (Dual-tone Multi-frequency)
    
    
    
    
    

    
    
    

  • DC removal

    • a DC-balanced signal has zero sum and its DTFT value at zero is zero.

    • We want to remove the DC bias from a non zero-centered signal

    • This means we want to kill the frequency component at omega = 0.

    • There is a simple way to do this.
      
      
      
      This method gives eliminates constant offset only.

    • For the digial high-pass filter,
      
      
      
      When you apply to the frequency response, this methods shows attenuation over the entire frequency domain. z

      To improve design (DC notch), put in a pole close to the one and inside of the circle for the stability.
      
      
      
      

    • Hum removal: Similar to DC removal but we want to remove a specific nonzero frequency.

• Filter design

  • Difficulties

    • Frequency response: passbands and stopbands

    • Phase: overall delay, linearity

    • some limits on computational resources and/or numerical precision.

  • Practical Limitation

    • smaller transition band, smaller error tolerance => higher filter order => more expensive, larger delay

  • IIR

    • Pros:

      • Computationally efficient

      • strong attenuation easy

      • good for audio

    • Cons:

      • stability issues

      • difficult to design for arbitrary response

      • nonlinear phase

  • FIR

    • Pros:

      • always stable

      • optimal design techniques exist

      • can be design with linear phase

    • Cons:

      • computationally much more expensive

      • may "sound" harsh