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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY
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Page-1

- Introduction - ( 2 - 2 )
- Sampling theory - ( 3 - 13 )
- Discrete-time signals - ( 14 - 25 )
- Discrete-time Systems - ( 26 - 37 )
- MATLAB (Linear Convolution) - ( 38 - 41 )
- Causality - ( 42 - 55 )
- Fourier analysis of discrete-time signals and systems - ( 56 - 78 )
- The z-transform and realization of digital filters - ( 79 - 159 )
- The Lattice structure- Introduction - ( 160 - 162 )
- Discrete Fourier series - ( 163 - 199 )
- Filtering through DFT/FFT - ( 200 - 215 )
- Fast Fourier transform - ( 216 - 232 )
- IIR Digital Filters - ( 233 - 258 )
- Digital filter design- The Butterworth filter - ( 259 - 287 )
- FIR digital filters - ( 288 - 369 )
- Multirate DSP - ( 370 - 390 )
- Up-sampling - ( 391 - 427 )
- FINITE WORD LENGTH EFFECTS - ( 428 - 428 )
- LIMIT CYCLES - ( 429 - 445 )

Topic:

LECTURE NOTES
ON
DIGITAL SIGNAL PROCESSING
IV B.Tech I semester (JNTUH-R13)
ELECTRICAL AND ELECTRONICS ENGINEERING
1

Digital Signal Processing – 1
I. Introduction
Introduction to Digital Signal Processing: Discrete time signals & sequences, Linear shift
invariant systems, Stability, and Causality, Linear constant coefficient difference equations,
Frequency domain representation of discrete time signals and systems.
Contents:
Sampling theory
Discrete-time signals
Transformation of the independent variable
Discrete-time systems
Linear constant coefficient difference equations
Fourier analysis of discrete-time signals and systems
Frequency response of discrete-time system
Properties of the discrete-time Fourier transform (DTFT)
1 of 77
2

Sampling theory
Illustrative example A continuous-time random signal is shown. Based on this several
important concepts are shown below. The signal is a continuous-time signal with continuous
amplitude. Such a signal is also called an analog signal.
x(t)
+
t
0
–
4
3
2
1
0
–1
–2
–3
–4
x(t)
8
7
6
5
4
3
2
1
0
nT
0
n
0
x(n)
5.5
5
Time
1T
1
2.8
2T
2
3.8
3T
3
5.3
4T
4
1.5
5T
5
4.6
6T
6
8.4
2
3
5
1
4
7
7T
7
6.9
8T
8
7.3
Time
Sampled signal. Discrete-time
signal – time is discrete,
amplitude is continuous.
Quantized. Quantization noise
6
7
(error). Digital signal – both
time and amplitude are discrete.
111 110 111 Encoded to 3 bits/sample.
Note this particular point exhibits saturation (out of
range). Rounded down to 7, not 8.
101 010 011 101 001 100
If we were to represent every sample value with infinite precision (for example, x(1) = 2.8--,
instead of being approximated as 2 or 3) then we would need registers and memory words of
arbitrarily large size. However, owing to a finite word length we round off the sample values (in
this case x(1) = 2.8-- will be rounded to 2). This introduces quantization noise or error.
2 of 77
3

The procedure of generating a discrete-time signal from an analog signal is shown in the
following block diagram. In the digital signal processing course we are mostly dealing with
discrete–time rather than digital signals and systems, the latter being a subset of the former.
Sample
(& Hold)
Continuous-time,
continuous amplitude
(Analog signal)
Quantizer
Discrete-time,
continuous
amplitude
Encoder
Discrete-time,
discrete amplitude
(Digital signal)
Encoded
digital
signal
The three boxes shown above can be represented by an analog to digital converter
(ADC). A complete digital signal processing (DSP) system consists of an ADC, a DSP algorithm
(e.g., a difference equation) and a digital to analog converter (DAC) shown below.
x(t)
x(n)
ADC
Algorithm
(Diff. Eq. or
y(n)
y(t)
DAC
equivalent)
As the name implies discrete-time signals are defined only at discrete instants of time.
Discrete-time signals can arise by sampling analog signals such as a voice signal or a
temperature signal in telemetry. Discrete-time signals may also arise naturally, e.g., the number
of cars sold on a specific day in a year, or the closing DJIA figure for a specific day of the year.
AT&T’s T1 Stream The voice signal is band limited to 3.3 kHz, sampled at 8000 Hz (8000
samples per second), quantized and encoded into 8 bits per sample. Twenty four such voice
channels are combined to form the T1 stream or signal.
1
= 0.125 msec.
Sampling interval =
8000 Hz
samples
bits
Bit rate for each channel = 8000
x8
= 64000 bits/sec.
sec
sample
Bit rate for T1 = 64000 bits/sec per channel x 24 channels = 1 544 000 bits/sec.
Commercial examples CD, Super Audio CD (SACD), DVD Audio (DVD-A), Digital audio
broadcasting - 32 kHz, and Digital audio tape (DAT) - 48 kHz.
Commercial Audio Examples
CD
Sampling
Rate
44.1 kHz
Super Audio CD
(SACD)
2.8224 MHz
Coding
16-bit PCM per sample
With 2 channels the bit rate is 1.4112
Mbits/sec, but additional error control
bits etc., raise it to 4.3218 Mbits/sec.
1-bit DSD
(Direct Stream
Digital – like Delta
modulation)
3 of 77
4
DVD-Audio
(DVD-A)
44.1, 88.2 or
48, 96, 192
kHz
12-, 20-, 24bit

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