As the speed of DSP's and digital hardware keeps increasing, software and digital hardware are replacing traditional analog hardware, making today's gizmos and gadgets smarter, more reliable, less expensive, and more power efficient than ever before. As an embedded systems engineer, you may have found yourself doing more signal processing than you originally bargained for! This article about complex notation is a survival guide to some of the most important basics of signal processing.

 

Begin reading nearly any article on signal processing and you will quickly encounter complex notation. Signals that could be easily represented as sines or cosines become Re(e^{×j(Δωt-Φ)}) and we're often reminded that j is the square root of -1. It gets worse Ð a simple sinusoid actually contains both positive and negative frequencies, but frequently "only positive frequencies are used." We also see modulated signals referred to as "complex envelopes." At first glance this notation seems like an unnecessary complication rather than an aid, and has caused many a good engineer to walk away from valuable signal processing information.

 

Most signal processing textbooks provide an excellent review of complex notation, but these descriptions are sometimes more general and involved than the working engineer wants. This article presents the basics of complex numbers as used in signal processing, and will provide you with the tools for using them. Through the examples used in this article you will gain a clear understanding of AC circuit analysis using phasors, and also get introduced to some of the key components of a QPSK receiver.

 

 

THE BASICS OF COMPLEX NUMBERS

Figure 1 shows a complex number. The "real" part of the number, Re(C), is represented by the horizontal component and the "imaginary" part of the number, Im(C), is represented by the vertical component. The complex number appears immediately useful because, like a sine wave, it has both a magnitude and an angle. If we change the sign of the imaginary part we obtain the complex conjugate of the number, and we designate the complex conjugate with the overbar symbol.

 

Figure 1- Complex number

 

The complex number can be represented various ways:

 

 

                                                                                                Equation 1           

                                                                                         

 

                                                            Equation 2           

 

The Euler identity shown below forms the basis of complex notation.

 

                                                                                              Equation 3                                                                 

 

Note that the rectangular forms of equations 1 and 2 allow convenient addition and subtraction of complex numbers. The polar form of equation 3 allows convenient multiplication and division.

 

If Φ is a phase angle which increases linearly with time then the complex number in Figure 1 rotates in a counter-clockwise circle. Letting the magnitude of the complex number be unity, equation 2 becomes

 

                                                                  Equation 4

 

and it's evident that the real part of this expression is a simple cosine wave. Furthermore, equations 1 through 4 can be easily manipulated to yield other useful forms

 

 

                                                              Equation 5

 

                                          Equation 6

 

                                                     Equation 7

 

                                                           Equation 8

 

Several things are readily apparent from equations 5 through 8. First they are related - equation 7 is the result of adding equations 5 and 6, and equation 8 is the result of subtracting equation 6 from equation 5. Equations 5 and 6 can also be viewed as two counter-rotating vectors in which the complex parts always cancel thereby accounting for the real result. Equation 6 can be viewed as two counter-rotating vectors where the real parts cancel. Equations 1 through 8 can also be used to derive useful identities such as:

 

                                     Equation 9

 

                                            Equation 10

 

But the most significant observation is that if we wish to take advantage of complex exponentials, we must either use the abstraction of complex numbers (equations 5 and 6) or the abstraction of negative frequencies (equations 7 and 8).

 

 

LINEAR FILTERING AND PHASORS

 

The problem of determining the steady state response of a linear network to a sinusoidal input is an excellent application of complex notation. However, working the problem once will show that steady-state analysis can be done virtually by inspection using the phasor technique. This section will reinforce what we've covered so far by showing why the phasor technique works.

 

Digital and analog filters can be described by complex frequency domain transfer functions. For example the lowpass RC filter shown in Figure 2 has the transfer function:

 

 

               Equation 11

 

 

This transfer function can be evaluated at different frequencies. At DC (w=0) the result is 1 + j0. At w=1/RC the result is 0.707 - j0.707, at w=´ the result is 0.

 

 

Figure 2 - RC filter

 

 

A property of this filter is that its transfer function has conjugate odd symmetry. This means that Equations 7 and 8 show that the sinusoidal excitation also has odd symmetry.

 

We will use the superposition property of linear networks to determine the output of the filter. Stated mathematically:

 

                                                                     Equation 12

 

In other words, the output due to the sum of two input signals is equal to the sum of the outputs due to the individual inputs. A quick glance at equations 5 through 8 suggests that this property should come in handy.

 

The input to the network is the cosine from equation 7. Using superposition, the output signal is simply the sum of the outputs due to the two components in equation 7.

 

                                             Equation 13

 

Applying the identity in equation 10 yields:

 

                                           Equation 14

 

Then apply the identity from equation 9:

 

                                                               Equation 15

 

 

                                          Equation 16

 

                                             Equation 17

 

                                        Equation 18

 

Because of the conjugate symmetry of both the input signal and the transfer function, the operations in equations 13 through 18 are identical for any network. As a result, to find the response of a network due to a sine wave, you simply need to know the magnitude and phase response of the network at the desired frequency. In circuit analysis complex numbers are represented using shorthand called phasor notation as shown below.

 

                                                                                  Equation 19

 

Setting the angular frequency to 1/RC, the steady state problem above is solved quickly using phasors as follows:

 

                                     Equation 20

 

                                                        Equation 21

 

 

 

 

 

NARROWBAND SIGNALS AND COMPLEX NOTATION

In signal processing we frequently work with narrowband signals as shown in Figure 3. A narrowband signal has its energy concentrated around a frequency usually near the center of the signal's bandwidth called the carrier. Channelized signals can usually be treated as narrowband signals. Complex notation allows us to focus only on the bandwidth containing the signal rather than the bandwidth containing both the signal and the carrier. It also suggests hardware and software structures that aid processing of lowpass signals.

 

 

 

Figure 3 - Narrowband signal

 

 

To illustrate, consider the signal processing used for QPSK data transmission shown in Figure 4.

 

 

 

Figure 4 - QPSK transmission system (click to zoom).

 

 

QPSK, or Quadrature Phase Shift Keying is a data transmission technique where groups of two bits of data are encoded into one of four phase shifts of a sinusoidal carrier. The phase shift is held for the symbol time, and then it is updated based on the next pair of bits in the input bit stream. Since the symbol rate is usually much lower than the carrier frequency, the result is a narrowband signal.

It is natural to use complex notation to represent the two bits comprising each symbol. The index k is used to indicate that the signal represents a pair of bits in a serial bit stream. Signals s_ik and s_qk are the k_th in phase and quadrature signal components that take values of ±1, and s_k denotes the complex signal s_ik + j s_qk. The complex baseband signal could also be represented as e^j×fk where f_k is 45, 135, 225, or 315 degrees. Mapping these signals on the complex plane results in the constellation shown in Figure 5.

 

 

Figure 5 - QPSK constellation

 

The two multipliers at the left of Figure 4 up-convert the complex baseband signal to the carrier frequency. This is shown graphically in Figure 6 where multiplication by e^j×w×t shifts the complex baseband spectrum to the right, and taking the real part adds the negative frequency component.

 

Figure 6 - Up-converter operations

 

 

Equations 22 to 25 describe the operation of the up-converter. The output is the real modulated signal m(t)[1].

 

                                                                   Equation 22

 

                          Equation 23

 

                Eq. 24

 

                                                            Equation 25

 

 

At the receiver the signal encounters a bandpass filter used for rejecting out-of-band signals. The bandpass filter response is not symmetric about the carrier frequency and it distorts the signal slightly.

 

The two multipliers and lowpass filters after the bandpass filter are used to down convert the signal to its lowpass representation. We'll elaborate on the down converter since it is a commonly encountered signal processing operation. One way to explain the operation of the down converter is to begin with the signal as described by equation 25, multiply the sines and cosines and note that the lowpass filter removes the double frequency terms resulting from the multiplications. But instead of using sines and cosines, we'll use our newfound skills with complex notation.

 

For the top arm we have:

 

                                                           Equation 26

 

The identities in equations 9 and 10 and the Euler identities allow us to write this as:

 

                            Equation 28

 

 

        Equation 29

 

Where:

 

                                       Equation 30

 

The lowpass filter removes the double frequency term on the right. Multiplying by 2 yields the final result:

 

                                                                                             Equation 31

 

Similarly the result for the bottom arm is:

 

                                                                                             Equation 32

 

 

The I and Q branches at the output of the down-converter are lowpass signals and can be sampled at a much lower rate than the modulated signal, which reduces both power consumption and hardware cost. Figure 4 shows the I and Q branches combined into a single heavy line indicating that the signal is complex. A clear advantage of complex notation is that the frequency and phase differences between the transmit and receive oscillators are maintained by equations 31 and 32.

 

The next step for the QPSK receiver to recover the transmitted data bits s_i and s_q is to correct for the frequency offset (Δω) and phase offset (Δf) between transmitter and receiver. The process of removing these offsets is called carrier recovery, and the receiver has the means to detect them and internally generate the signal e^{×j(Δωt-Φ)} The process of applying the correction is often referred to as de-rotation because the frequency offset causes the baseband phasor to rotate at the difference frequency. However this is simply complex multiplication as shown by equation 33 and implemented in Figure 7. Though the structure appears formidable, it can be done in just a few lines of DSP code.

 

 

 

        Equation 33

 

 

 

 

Figure 7 - Complex multiplier

 

 

The final step f