Sampling theory Assignment Help

Digital signal processing - Sampling theory

Sampling theory

A continuous time random signal is shown. Based on this a number of important concepts are shown below. The signal is continuous time signal with continuous amplitude. This type of signal is known as an analog signal.

1884_sampling theory.png

 

 

 

 

 

 

 

 

 

 

 

 

 

x(t)↑

 

 

 

 

 

 

 

 

 

 

4

8

 

 

 

 

 

 

 

 

 

 

3

7

 

 

 

 

 

 

 

 

 

 

2

6

 

 

 

 

 

 

 

 

 

 

1

5

 

 

 

 

 

 

 

 

 

 

0

4

 

 

 

 

 

 

 

 

 

→Time

-1

3

 

 

 

 

 

 

 

 

 

 

-2

2

 

 

 

 

 

 

 

 

 

 

-3

1

 

 

 

 

 

 

 

 

 

 

-4

0

 

 

 

 

 

 

 

 

 

 

nT→

0

1T

2T

3T

4T

5T

6T

7T

8T

→Time

n→

0

1

2

3

4

5

6

7

8

 

x(n)→

5.5

2.8

3.8

5.3

1.5

4.6

8.4

6.9

7.3

←Sampled signal. Discrete-time

signal - time is discrete, amplitude is continuous.

 

 

5

2

3

5

1

4

7

6

7

←Quantized. Quantization noise

(error). Digital signal - both time and amplitude are discrete.

 

 

101

010

011

101

001

100

111

110

111

Encoded to 3 bits/sample.

 

 

 

 

 

 

 

 

↑Note this particular point exhibits saturation (out of

range). Rounded down to 7, not 8.

 

If we were to represent sample value with infinite precision then we would require registers and memory words of arbitrarily large size. But, because of a finite word length we round off the sample values. This introduces the quantization noise.

The process of generating a discrete-time signal from the analog signal is shown in block diagram. In the digital signal processing course we are mostly dealing with discrete-time rather than digital signals and systems, the latter is a subset of the former.

2044_sampling theory1.png

The 3 boxes shown in the above diagram can be represented by an analog to digital converter (ADC). A complete digital signal processing (DSP) system comprises of an ADC, a DSP algorithm

(for example a difference equation) and a digital to analog converter (DAC) which is shown below.

1475_sampling theory2.png

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.

Sampling interval = 1/8000 Hz = 0.125 msec.

Bit rate for each channel = 8000 samples/sec  x 8 bits/sample = 64000 bits/sec.

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

Super Audio CD

(SACD)

DVD-Audio

(DVD-A)

Sampling

Rate

44.1 kHz

2.8224 MHz

44.1, 88.2 or

48, 96, 192 kHz

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)

12-, 20-, 24- bit

 

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