Introduction to DSP Systems Dr. Deepa Kundur University of Toronto Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 1 / 30 Analog vs. Digital Analog and Digital Signals I analog signal = continuous-time + continuous amplitude I digital signal = discrete-time + discrete amplitude t 2 1 2 3 -1 -2 -3 4 -2 -4 x(t) t 1 2 1 2 3 -1 -2 -3 0.5 1.5 2.5 4 0.5 -2 -4 x(t) -1 1 0 n x[n] -2 -3 2 3 1 -1 1 0 n x[n] -2 -3 2 3 2 1 1 continuous-time discrete-time continuous amplitude discrete amplitude Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 2 / 30 Analog vs. Digital Analog and Digital Signals I Analog signals are fundamentally significant because we must interface with the real world which is analog by nature. I Digital signals are important because they facilitate the use of digital signal processing (DSP) systems, which have practical and performance advantages for several applications. Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 3 / 30 Analog vs. Digital Analog and Digital Systems I analog system = analog signal input + analog signal output I advantages : easy to interface to real world, do not need A/D or D/A converters, speed not dependent on clock rate I digital system = digital signal input + digital signal output I advantages : re-configurability using software, greater control over accuracy/resolution, predictable and reproducible behavior Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 4 / 30
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Introduction to DSP Systems
Dr. Deepa Kundur
University of Toronto
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 1 / 30
Analog vs. Digital
Analog and Digital Signals
I analog signal = continuous-time + continuous amplitude
I digital signal = discrete-time + discrete amplitude
t
2
1 2 3-1-2-3 4
-2
-4
x(t)
t1
2
1 2 3-1-2-3 0.5 1.5 2.5 4
0.5
-2
-4
x(t)
-1 10n
x[n]
-2-3 2 3
1
-1 10n
x[n]
-2-3 2 3
2
1 1
continuous-time
discrete-time
continuous amplitude discrete amplitude
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 2 / 30
Analog vs. Digital
Analog and Digital Signals
I Analog signals are fundamentally significant because we mustinterface with the real world which is analog by nature.
I Digital signals are important because they facilitate the use ofdigital signal processing (DSP) systems, which have practicaland performance advantages for several applications.
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 3 / 30
Analog vs. Digital
Analog and Digital Systems
I analog system =analog signal input + analog signal output
I advantages: easy to interface to real world, do not need A/D orD/A converters, speed not dependent on clock rate
I digital system =digital signal input + digital signal output
I advantages: re-configurability using software, greater controlover accuracy/resolution, predictable and reproducible behavior
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 4 / 30
Analog vs. Digital
Analog-to-Digital Conversion
CoderQuantizerx(n)x (t)a x (n)q
Analogsignal
Discrete-timesignal
Quantizedsignal
Digitalsignal
A/D converter
0 1 0 1 1 . . .Sampler
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 5 / 30
Analog vs. Digital
Analog-to-Digital Conversion
CoderQuantizerx(n)x (t)a x (n)q
Analogsignal
Discrete-timesignal
Quantizedsignal
Digitalsignal
A/D converter
0 1 0 1 1 . . .Sampler
Sampling:I conversion from cts-time to dst-time by taking “samples” at
discrete time instants
I E.g., uniform sampling: x(n) = xa(nT ) where T is the samplingperiod and n ∈ Z
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 6 / 30
Analog vs. Digital
Analog-to-Digital Conversion
CoderQuantizerx(n)x (t)a x (n)q
Analogsignal
Discrete-timesignal
Quantizedsignal
Digitalsignal
A/D converter
0 1 0 1 1 . . .Sampler
Sampling:
-1 1 0 n
x[n]
-2 -3 2 3
1
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 7 / 30
Analog vs. Digital
Analog-to-Digital Conversion
CoderQuantizerx(n)x (t)a x (n)q
Analogsignal
Discrete-timesignal
Quantizedsignal
Digitalsignal
A/D converter
0 1 0 1 1 . . .Sampler
Quantization:I conversion from dst-time cts-valued signal to a dst-time
dst-valued signal
I quantization error: eq(n) = xq(n)− x(n) for all n ∈ Z
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 8 / 30
Analog vs. Digital
Analog-to-Digital Conversion
CoderQuantizerx(n)x (t)a x (n)q
Analogsignal
Discrete-timesignal
Quantizedsignal
Digitalsignal
A/D converter
0 1 0 1 1 . . .Sampler
Quantization:
-1 1 0 n
x [n]
-2 -3 2 3
1
q
01234567
01234567
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 9 / 30
Analog vs. Digital
Analog-to-Digital Conversion
CoderQuantizerx(n)x (t)a x (n)q
Analogsignal
Discrete-timesignal
Quantizedsignal
Digitalsignal
A/D converter
0 1 0 1 1 . . .Sampler
Coding:I representation of each dst-value xq(n) by a
b-bit binary sequence
I e.g., if for any n, xq(n) ∈ {0, 1, . . . , 6, 7}, then the coder mayuse the following mapping to code the quantized amplitude:
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 10 / 30
Analog vs. Digital
Analog-to-Digital Conversion
CoderQuantizerx(n)x (t)a x (n)q
Analogsignal
Discrete-timesignal
Quantizedsignal
Digitalsignal
A/D converter
0 1 0 1 1 . . .Sampler
Example coder:
0 000 4 1001 001 5 1012 010 6 1103 011 7 111
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 11 / 30
Analog vs. Digital
Sampling Theorem
If the highest frequency contained in an analog signal xa(t) isFmax = B and the signal is sampled at a rate
Fs > 2Fmax = 2B
then xa(t) can be exactly recovered from its sample values using theinterpolation function
g(t) =sin(2πBt)
2πBt
Note: FN = 2B = 2Fmax is called the Nyquist rate.
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 12 / 30
Analog vs. Digital
Sampling Theorem
Sampling Period = T =1
Fs=
1
Sampling Frequency
Therefore, given the interpolation relation, xa(t) can be written as
xa(t) =∞∑
n=−∞
xa(nT )g(t − nT )
xa(t) =∞∑
n=−∞
x(n) g(t − nT )
where xa(nT ) = x(n); called bandlimited interpolation.
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 13 / 30
Analog vs. Digital
Digital-to-Analog Conversion
-1 1 0 n
x[n]
-2 -3 2 3
1
original/bandlimitedinterpolated signal
I Common interpolation approaches: bandlimited interpolation,zero-order hold, linear interpolation, higher-order interpolationtechniques, e.g., using splines
I In practice, “cheap” interpolation along with a smoothing filteris employed.
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 14 / 30
Analog vs. Digital
Digital-to-Analog Conversion
-T T0 t
-2T 2T 3T
1
original/bandlimitedinterpolated signal
zero-orderhold
-3T
I Common interpolation approaches: bandlimited interpolation,zero-order hold, linear interpolation, higher-order interpolationtechniques, e.g., using splines
I In practice, “cheap” interpolation along with a smoothing filteris employed.
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 15 / 30
Analog vs. Digital
Digital-to-Analog Conversion
-T T0 t
-2T 2T 3T
1
-3T
linearinterpolation
original/bandlimitedinterpolated signal
I Common interpolation approaches: bandlimited interpolation,zero-order hold, linear interpolation, higher-order interpolationtechniques, e.g., using splines
I In practice, “cheap” interpolation along with a smoothing filteris employed.
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 16 / 30
DSP Systems
A DSP System
A/D DSP D/A
Analogsignal
Analogsignal
Sampled datasignal
Analogsignal
Cts-time dst-amp “staricase” signal
Digitalsignal
Digitalsignal
DSP System
AntialiasingFilter
Sample and Hold
Reconstruction
Filter
I In practice, a DSP system does not use idealized A/D or D/Amodels.
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 17 / 30
DSP Systems
A DSP System
A/D DSP D/A
Analogsignal
Analogsignal
Sampled datasignal
Analogsignal
Cts-time dst-amp “staricase” signal
Digitalsignal
Digitalsignal
DSP System
AntialiasingFilter
Sample and Hold
Reconstruction
Filter
Anti-aliasing Filter:I ensures that analog input signal does not contain frequency
components higher than half of the sampling frequency (to obeythe sampling theorem)
I this process is irreversible
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 18 / 30
DSP Systems
A DSP System
t
2
1 2 3-1-2-3 4
-2
-4
t1
2
1 2 3-1-2-3 0.5 1.5 2.5 4
0.5
-2
-4
x(t)
-1 10n
x[n]
-2-3 2 3
1
-1 10n
x[n]
-2-3 2 3
2
1 1
Input Signal
t
2
1 2 3-1-2-3 4
-2
-4
t1
2
1 2 3-1-2-3 0.5 1.5 2.5 4
0.5
-2
-4
x(t)
-1 10n
x[n]
-2-3 2 3
1
-1 10n
x[n]
-2-3 2 3
2
1 1
Anti-aliased Signal
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 19 / 30
DSP Systems
A DSP System
A/D DSP D/A
Analogsignal
Analogsignal
Sampled datasignal
Analogsignal
Cts-time dst-amp “staricase” signal
Digitalsignal
Digitalsignal
DSP System
AntialiasingFilter
Sample and Hold
Reconstruction
Filter
Sample and Hold:I holds a sampled analog value for a short time while the A/D
converts and interprets the value as a digital
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 20 / 30
DSP Systems
A DSP System
t
2
1 2 3-1-2-3 4
-2
-4
t1
2
1 2 3-1-2-3 0.5 1.5 2.5 4
0.5
-2
-4
x(t)
-1 10n
x[n]
-2-3 2 3
1
-1 10n
x[n]
-2-3 2 3
2
1 1
Anti-aliased Signal
t
2
1 2 3-1-2-3 4
-2
-4
t1
2
1 2 3-1-2-3 0.5 1.5 2.5 4
0.5
-2
-4
x(t)
-1 10n
x[n]
-2-3 2 3
1
-1 10n
x[n]
-2-3 2 3
2
1 1
Sampled Data Signal
anti-aliased signal
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 21 / 30
DSP Systems
A DSP System
A/D DSP D/A
Analogsignal
Analogsignal
Sampled datasignal
Analogsignal
Cts-time dst-amp “staricase” signal
Digitalsignal
Digitalsignal
DSP System
AntialiasingFilter
Sample and Hold
Reconstruction
Filter
A/D:I converts a sampled data signal value into a digital number, in
part, through quantization of the amplitude
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 22 / 30
DSP Systems
A DSP System
t
2
1 2 3-1-2-3 4
-2
-4
t1
2
1 2 3-1-2-3 0.5 1.5 2.5 4
0.5
-2
-4
x(t)
-1 10n
x[n]
-2-3 2 3
1
-1 10n
x[n]
-2-3 2 3
2
1 1
Sampled Data Signal
anti-aliased signal
t
2
1 2 3-1-2-3 4
-2
-4
t1
2
1 2 3-1-2-3 0.5 1.5 2.5 4
0.5
-2
-4
x(t)
-1 10n
x[n]
-2-3 2 3
1
-1 10n
x[n]
-2-3 2 3
2
1 1
Digital Signal
sampled data signal
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 23 / 30
DSP Systems
A DSP System
A/D DSP D/A
Analogsignal
Analogsignal
Sampled datasignal
Analogsignal
Cts-time dst-amp “staricase” signal
Digitalsignal
Digitalsignal
DSP System
AntialiasingFilter
Sample and Hold
Reconstruction
Filter
D/A:I converts a digital signal into a “staircase”-like signal
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 24 / 30
DSP Systems
A DSP System
t
2
1 2 3-1-2-3 4
-2
-4
t1
2
1 2 3-1-2-3 0.5 1.5 2.5 4
0.5
-2
-4
x(t)
-1 10n
x[n]
-2-3 2 3
1
-1 10n
x[n]
-2-3 2 3
2
1 1
Digital Signal
sampled data signal
t
2
1 2 3-1-2-3 4
-2
-4
t1
2
1 2 3-1-2-3 0.5 1.5 2.5 4
0.5
-2
-4
x(t)
-1 10n
x[n]
-2-3 2 3
1
-1 10n
x[n]
-2-3 2 3
2
1 1
Staircase Signaldigital signal
sampled data signal
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 25 / 30
DSP Systems
A DSP System
A/D DSP D/A
Analogsignal
Analogsignal
Sampled datasignal
Analogsignal
Cts-time dst-amp “staricase” signal
Digitalsignal
Digitalsignal
DSP System
AntialiasingFilter
Sample and Hold
Reconstruction
Filter
Reconstruction Filter:I converts a “staircase”-like signal into an analog signal through
lowpass filtering similar to the type used for anti-aliasing
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 26 / 30
DSP Systems
A DSP System
t
2
1 2 3-1-2-3 4
-2
-4
t1
2
1 2 3-1-2-3 0.5 1.5 2.5 4
0.5
-2
-4
x(t)
-1 10n
x[n]
-2-3 2 3
1
-1 10n
x[n]
-2-3 2 3
2
1 1
Staircase Signaldigital signal
sampled data signal
t
2
1 2 3-1-2-3 4
-2
-4
t1
2
1 2 3-1-2-3 0.5 1.5 2.5 4
0.5
-2
-4
x(t)
-1 10n
x[n]
-2-3 2 3
1
-1 10n
x[n]
-2-3 2 3
2
1 1
Reconstructed Signal
anti-aliased signal
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 27 / 30
DSP Systems
Real-time DSP Considerations
Q: What are initial considerations when designing a DSP system thatmust run in real-time?
I Algorithm: related to computational operations and accuracyrequired by the application
I Sample rate: the rate at which input samples are received forprocessing
I Speed: to meet an application throughput requirement with agiven sample rate, it must be possible to operate the DSP at aparticular speed
I Numeric representation: format and number of bits used fordata representation; depends on required computationalprecision and dynamic range required for application
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 28 / 30
DSP Systems
Real-time DSP Considerations
Q: Is a DSP technology suitable for a real-time application?
I Clock rate: rate at which a DSP performs its most basic unit of
work; to meet the timing requirement with a given sampling rate, it
must be possible to operate the DSP at a particular clock rate
I Throughput: rate of multiply and accumulates (MACs) performed;
measured in number of MACs per second
I Arithmetic and addressing capability: requirements related to the
algorithm complexity, precision and data access
I Precision: associated with format (fixed vs. floating), number of bits
used for data representation, and required dynamic range
I Size, cost and power consumption: technology-dependent
Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 29 / 30
DSP Systems
Programmable DSPs
I Application-specific: designed to perform one function moreaccurately, faster or more cost-effectively
I examples: FFT chips, digital filtersI can be programmable within confines of a function; e.g.,
coefficients of a digital filter
I General purpose: microprocessor whose architecture is optimizedto process sampled data at high rates via pipelining andparallelism
I programmable and more cost-effective for general computingI short system design cycle time
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Dr. Deepa Kundur (University of Toronto) Introduction to DSP Systems 30 / 30