C55x_Course_11
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NCTU EECS DSP Lab
11-4
• Frequency quantization in finite word length
cos-1(a + q) - cos-1(a) = δω, a = cos(ω) q : quantum size (2-15 = 3.052×10-5) Frequency quantization: 0.0078 ≤ δω ≤ 3.052×10-5 – What is the frequency quantization in analog domain? – Can we improve the accuracy by increasing the sampling rate?
• Generating arbitrary narrow-band wave
y(t) = a0 + a1cos(2πf1t) + b1sin(2πf1t) + a2cos(2πf2t) + b2sin(2πf2t) + a3cos(2πf3t) + b3sin(2πf3t) + … yd(k) = a0 + a1cos(2πkf1T) + b1sin(2πkf1T) + a2cos(2πkf2T) + b2sin(2πkf2T) + … – What is the limitation of yd(k)?
• Sinusoidal wave form
– Table look-up (requiring some memory space) • Good precision for fixed frequency • Variable frequency – Changing sampling rate – Using interpolation + decimation • Difficult to perform in real-time • Phase control
conditions
Impulse input : u(0) = 1 , u(k) = 0 for k ≠ 0 • Sin wave : c0 = 0, c1 = A sin(ω) • Cos wave: c0 = A, c1 = −A cos(ω) • Arbitrary phase : c0 = A sin(φ), c1 = A sin(ω−φ)
y ( 0 ) = A sin( φ ), y (1) = A sin( ω + φ )
NCTU EECS DSP Lab
11-3
Frequency limitation
ω 1 f = < or 0 ≤ ω ≤ π 2π T 2T
• Using impulse response y(k) = c0 + c1 q−1 1 − 2cos(ω) q−1 + q−2 u(k) Zero initial
An all-pole IIR filter
y ( k ) = 2 cos( ω ) y ( k − 1) − y ( k − 2 )
Initial conditions for sin wave
y ( 0 ) = 0 , y (1) = A sin( ω )
Initial conditions for cos wave y ( 0 ) = A , y (1) = A cos( ω ) Initial conditions for arbitrary phase
NCTU EECS DSP Lab
11-2
• Generating sinusoidal wave using IIR filters
– Single frequency case
[cos(ωk )] = 2 cos(ω )[cos(ω (k −1))] − [cos(ω (k − 2))] [sin(ωk )] = 2 cos(ω )[sin(ω (k −1))] − [sin(ω (k − 2))]
NCTU EECS DSP Lab
11-6
Lab DTMF
• DTMF (Dial Tone Multiple Frequency)
NCTU EECS DSP Lab
11-7
DTMF Specification
• Tby AT&T state the following:
– 10 digits/sec are the maximum data rate for touch tone signals. – For a 100 msec time slot the duration for the actual tone is at least 45 msec and not longer than 55 msec. – The tone generator must be quiet during the remainder of the 100 msec time slot.
11-8
NCTU EECS DSP Lab
NCTU EECS DSP Lab
11-5
• Generating an arbitrary periodic wave
– Fundamental period = P
y(t) = a0 + a1cos(2πt/P) + b1sin(2πt/P) + a2cos(4πt/P) + b2sin(4πt/P) + a3cos(6πt/P) + b3sin(6πt/P) + … yd(k) = a0 + a1cos(2πkT/P) + b1sin(2πkT/P) + a2cos(4πkT/P) + b2sin(4πkT/P) + …
Lecture 11
• Lecture – Fast Sine wave generator • Lab – DTMF signal generation
NCTU EECS DSP Lab
11-1
Sinusoidal Waveform Generation
• Wave forms
– Sinusoidal, triangular, random, arbitrary
11-4
• Frequency quantization in finite word length
cos-1(a + q) - cos-1(a) = δω, a = cos(ω) q : quantum size (2-15 = 3.052×10-5) Frequency quantization: 0.0078 ≤ δω ≤ 3.052×10-5 – What is the frequency quantization in analog domain? – Can we improve the accuracy by increasing the sampling rate?
• Generating arbitrary narrow-band wave
y(t) = a0 + a1cos(2πf1t) + b1sin(2πf1t) + a2cos(2πf2t) + b2sin(2πf2t) + a3cos(2πf3t) + b3sin(2πf3t) + … yd(k) = a0 + a1cos(2πkf1T) + b1sin(2πkf1T) + a2cos(2πkf2T) + b2sin(2πkf2T) + … – What is the limitation of yd(k)?
• Sinusoidal wave form
– Table look-up (requiring some memory space) • Good precision for fixed frequency • Variable frequency – Changing sampling rate – Using interpolation + decimation • Difficult to perform in real-time • Phase control
conditions
Impulse input : u(0) = 1 , u(k) = 0 for k ≠ 0 • Sin wave : c0 = 0, c1 = A sin(ω) • Cos wave: c0 = A, c1 = −A cos(ω) • Arbitrary phase : c0 = A sin(φ), c1 = A sin(ω−φ)
y ( 0 ) = A sin( φ ), y (1) = A sin( ω + φ )
NCTU EECS DSP Lab
11-3
Frequency limitation
ω 1 f = < or 0 ≤ ω ≤ π 2π T 2T
• Using impulse response y(k) = c0 + c1 q−1 1 − 2cos(ω) q−1 + q−2 u(k) Zero initial
An all-pole IIR filter
y ( k ) = 2 cos( ω ) y ( k − 1) − y ( k − 2 )
Initial conditions for sin wave
y ( 0 ) = 0 , y (1) = A sin( ω )
Initial conditions for cos wave y ( 0 ) = A , y (1) = A cos( ω ) Initial conditions for arbitrary phase
NCTU EECS DSP Lab
11-2
• Generating sinusoidal wave using IIR filters
– Single frequency case
[cos(ωk )] = 2 cos(ω )[cos(ω (k −1))] − [cos(ω (k − 2))] [sin(ωk )] = 2 cos(ω )[sin(ω (k −1))] − [sin(ω (k − 2))]
NCTU EECS DSP Lab
11-6
Lab DTMF
• DTMF (Dial Tone Multiple Frequency)
NCTU EECS DSP Lab
11-7
DTMF Specification
• Tby AT&T state the following:
– 10 digits/sec are the maximum data rate for touch tone signals. – For a 100 msec time slot the duration for the actual tone is at least 45 msec and not longer than 55 msec. – The tone generator must be quiet during the remainder of the 100 msec time slot.
11-8
NCTU EECS DSP Lab
NCTU EECS DSP Lab
11-5
• Generating an arbitrary periodic wave
– Fundamental period = P
y(t) = a0 + a1cos(2πt/P) + b1sin(2πt/P) + a2cos(4πt/P) + b2sin(4πt/P) + a3cos(6πt/P) + b3sin(6πt/P) + … yd(k) = a0 + a1cos(2πkT/P) + b1sin(2πkT/P) + a2cos(4πkT/P) + b2sin(4πkT/P) + …
Lecture 11
• Lecture – Fast Sine wave generator • Lab – DTMF signal generation
NCTU EECS DSP Lab
11-1
Sinusoidal Waveform Generation
• Wave forms
– Sinusoidal, triangular, random, arbitrary
